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Theorem ptbasfi 21908
Description: The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add 𝑋 itself to the list because if 𝐴 is empty we get (fi‘∅) = ∅ while 𝐵 = {∅}.) (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
ptbasfi.2 𝑋 = X𝑛𝐴 (𝐹𝑛)
Assertion
Ref Expression
ptbasfi ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
Distinct variable groups:   𝑘,𝑛,𝑢,𝐵   𝑤,𝑔,𝑥,𝑦,𝑛,𝑘,𝑢,𝑧,𝐴   𝑔,𝐹,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧   𝑔,𝑋,𝑘,𝑢,𝑤,𝑥,𝑧   𝑔,𝑉,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑔)   𝑋(𝑦,𝑛)

Proof of Theorem ptbasfi
Dummy variables 𝑠 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptbas.1 . . . . 5 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
21elpt 21899 . . . 4 (𝑠𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)))
3 df-3an 1071 . . . . . . . 8 (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ↔ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
4 simprr 761 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))
5 disjdif2 4314 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) = ∅ → (𝐴𝑚) = 𝐴)
65raleqdv 3357 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) = (𝐹𝑦)))
76biimpac 471 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → ∀𝑦𝐴 (𝑦) = (𝐹𝑦))
8 ixpeq2 8279 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑦) = (𝐹𝑦) → X𝑦𝐴 (𝑦) = X𝑦𝐴 (𝐹𝑦))
97, 8syl 17 . . . . . . . . . . . . . . 15 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = X𝑦𝐴 (𝐹𝑦))
10 ptbasfi.2 . . . . . . . . . . . . . . . 16 𝑋 = X𝑛𝐴 (𝐹𝑛)
11 fveq2 6504 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑦 → (𝐹𝑛) = (𝐹𝑦))
1211unieqd 4727 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 (𝐹𝑛) = (𝐹𝑦))
1312cbvixpv 8283 . . . . . . . . . . . . . . . 16 X𝑛𝐴 (𝐹𝑛) = X𝑦𝐴 (𝐹𝑦)
1410, 13eqtri 2804 . . . . . . . . . . . . . . 15 𝑋 = X𝑦𝐴 (𝐹𝑦)
159, 14syl6eqr 2834 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
164, 15sylan 572 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = 𝑋)
17 ssv 3883 . . . . . . . . . . . . . . . 16 𝑋 ⊆ V
18 iineq1 4813 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
19 0iin 4858 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ∅ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V
2018, 19syl6eq 2832 . . . . . . . . . . . . . . . 16 ((𝐴𝑚) = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
2117, 20syl5sseqr 3912 . . . . . . . . . . . . . . 15 ((𝐴𝑚) = ∅ → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
2221adantl 474 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → 𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
23 df-ss 3845 . . . . . . . . . . . . . 14 (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2422, 23sylib 210 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
2516, 24eqtr4d 2819 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) = ∅) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
26 simplll 763 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝐴𝑉𝐹:𝐴⟶Top))
27 inss1 4095 . . . . . . . . . . . . . . . . 17 (𝐴𝑚) ⊆ 𝐴
28 simpr 477 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
2927, 28sseldi 3858 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝐴)
30 fveq2 6504 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝑦) = (𝑛))
31 fveq2 6504 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (𝐹𝑦) = (𝐹𝑛))
3230, 31eleq12d 2862 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝑛) ∈ (𝐹𝑛)))
33 simprr 761 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3433ad2antrr 714 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
3532, 34, 29rspcdva 3543 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛) ∈ (𝐹𝑛))
3614ptpjpre1 21898 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛))) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3726, 29, 35, 36syl12anc 825 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3837adantlr 703 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
3938iineq2dv 4821 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
40 simpr 477 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝐴𝑚) ≠ ∅)
41 cnvimass 5794 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ dom (𝑤𝑋 ↦ (𝑤𝑛))
42 eqid 2780 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) = (𝑤𝑋 ↦ (𝑤𝑛))
4342dmmptss 5939 . . . . . . . . . . . . . . . . . . . 20 dom (𝑤𝑋 ↦ (𝑤𝑛)) ⊆ 𝑋
4441, 43sstri 3869 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋
4544, 14sseqtri 3895 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
4645rgenw 3102 . . . . . . . . . . . . . . . . 17 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)
47 r19.2z 4326 . . . . . . . . . . . . . . . . 17 (((𝐴𝑚) ≠ ∅ ∧ ∀𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦)) → ∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
4840, 46, 47sylancl 578 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
49 iinss 4851 . . . . . . . . . . . . . . . 16 (∃𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
5048, 49syl 17 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ X𝑦𝐴 (𝐹𝑦))
5150, 14syl6sseqr 3910 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
52 sseqin2 4082 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋 ↔ (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5351, 52sylib 210 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
5433ad2antrr 714 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))
55 ssralv 3925 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ⊆ 𝐴 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦)))
5627, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦))
57 elssuni 4746 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ (𝐹𝑦))
58 iffalse 4362 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
5958sseq2d 3891 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 = 𝑛 → ((𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝑦) ⊆ (𝐹𝑦)))
6057, 59syl5ibrcom 239 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦) ∈ (𝐹𝑦) → (¬ 𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
61 ssid 3881 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦) ⊆ (𝑦)
62 iftrue 4359 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
6362, 30eqtr4d 2819 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
6461, 63syl5sseqr 3912 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑛 → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6560, 64pm2.61d2 174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6665ralrimivw 3135 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ∈ (𝐹𝑦) → ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
67 ssiin 4850 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑛 ∈ (𝐴𝑚)(𝑦) ⊆ if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6866, 67sylibr 226 . . . . . . . . . . . . . . . . . . . 20 ((𝑦) ∈ (𝐹𝑦) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
6968adantl 474 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) ⊆ 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
7062equcoms 1978 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑛))
71 fveq2 6504 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑦 → (𝑛) = (𝑦))
7270, 71eqtrd 2816 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑦 → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝑦))
7372sseq1d 3890 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) ↔ (𝑦) ⊆ (𝑦)))
7473rspcev 3537 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ⊆ (𝑦)) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7561, 74mpan2 679 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝐴𝑚) → ∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
76 iinss 4851 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7775, 76syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐴𝑚) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7877adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ⊆ (𝑦))
7969, 78eqssd 3877 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝐴𝑚) ∧ (𝑦) ∈ (𝐹𝑦)) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8079ralimiaa 3111 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (𝐴𝑚)(𝑦) ∈ (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
8154, 56, 803syl 18 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
82 eldifn 3996 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝐴𝑚) → ¬ 𝑦𝑚)
8382ad2antlr 715 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦𝑚)
84 inss2 4096 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑚) ⊆ 𝑚
85 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚))
8684, 85sseldi 3858 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → 𝑛𝑚)
87 eleq1 2855 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑛 → (𝑦𝑚𝑛𝑚))
8886, 87syl5ibrcom 239 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑦 = 𝑛𝑦𝑚))
8983, 88mtod 190 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → ¬ 𝑦 = 𝑛)
9089, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) ∧ 𝑛 ∈ (𝐴𝑚)) → if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = (𝐹𝑦))
9190iineq2dv 4821 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚) (𝐹𝑦))
92 iinconst 4807 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑚) ≠ ∅ → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9392adantr 473 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → 𝑛 ∈ (𝐴𝑚) (𝐹𝑦) = (𝐹𝑦))
9491, 93eqtr2d 2817 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
95 eqeq1 2784 . . . . . . . . . . . . . . . . . . 19 ((𝑦) = (𝐹𝑦) → ((𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (𝐹𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9694, 95syl5ibrcom 239 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴𝑚)) → ((𝑦) = (𝐹𝑦) → (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
9796ralimdva 3129 . . . . . . . . . . . . . . . . 17 ((𝐴𝑚) ≠ ∅ → (∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
984, 97mpan9 499 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
99 inundif 4313 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑚) ∪ (𝐴𝑚)) = 𝐴
10099raleqi 3355 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
101 ralunb 4058 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ ((𝐴𝑚) ∪ (𝐴𝑚))(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
102100, 101bitr3i 269 . . . . . . . . . . . . . . . 16 (∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦))))
10381, 98, 102sylanbrc 575 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → ∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
104 ixpeq2 8279 . . . . . . . . . . . . . . 15 (∀𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) → X𝑦𝐴 (𝑦) = X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
105103, 104syl 17 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
106 ixpiin 8291 . . . . . . . . . . . . . . 15 ((𝐴𝑚) ≠ ∅ → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
107106adantl 474 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 𝑛 ∈ (𝐴𝑚)if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
108105, 107eqtrd 2816 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = 𝑛 ∈ (𝐴𝑚)X𝑦𝐴 if(𝑦 = 𝑛, (𝑛), (𝐹𝑦)))
10939, 53, 1083eqtr4rd 2827 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ (𝐴𝑚) ≠ ∅) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
11025, 109pm2.61dane 3057 . . . . . . . . . . 11 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) = (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))))
111 ixpexg 8289 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑛𝐴 (𝐹𝑛) ∈ V → X𝑛𝐴 (𝐹𝑛) ∈ V)
112 fvex 6517 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹𝑛) ∈ V
113112uniex 7289 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹𝑛) ∈ V
114113a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐴 (𝐹𝑛) ∈ V)
115111, 114mprg 3104 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝐴 (𝐹𝑛) ∈ V
11610, 115eqeltri 2864 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ V
117116mptex 6818 . . . . . . . . . . . . . . . . . . . . 21 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
118117cnvex 7451 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝑋 ↦ (𝑤𝑛)) ∈ V
119118imaex 7442 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V
120119dfiin2 4834 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))}
121 inteq 4757 . . . . . . . . . . . . . . . . . 18 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅)
122120, 121syl5eq 2828 . . . . . . . . . . . . . . . . 17 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = ∅)
123 int0 4768 . . . . . . . . . . . . . . . . 17 ∅ = V
124122, 123syl6eq 2832 . . . . . . . . . . . . . . . 16 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) = V)
125124ineq2d 4079 . . . . . . . . . . . . . . 15 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = (𝑋 ∩ V))
126 inv1 4237 . . . . . . . . . . . . . . 15 (𝑋 ∩ V) = 𝑋
127125, 126syl6eq 2832 . . . . . . . . . . . . . 14 ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅ → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
128127adantl 474 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑋)
129 snex 5192 . . . . . . . . . . . . . . . . . 18 {𝑋} ∈ V
1301ptbas 21906 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
1311, 10ptpjpre2 21907 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
132131ralrimivva 3143 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵)
133 eqid 2780 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
134133fmpox 7579 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝐵 ↔ (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
135132, 134sylib 210 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
136135frnd 6356 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝐵)
137130, 136ssexd 5088 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V)
138 unexg 7295 . . . . . . . . . . . . . . . . . 18 (({𝑋} ∈ V ∧ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ∈ V) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
139129, 137, 138sylancr 579 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
140 ssfii 8684 . . . . . . . . . . . . . . . . 17 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
141139, 140syl 17 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
142141ad2antrr 714 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
143 ssun1 4039 . . . . . . . . . . . . . . . . 17 {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
144116snss 4597 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ↔ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
145143, 144mpbir 223 . . . . . . . . . . . . . . . 16 𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
146145a1i 11 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → 𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
147142, 146sseldd 3861 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
148147adantr 473 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
149128, 148eqeltrd 2868 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} = ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
150139ad3antrrr 718 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V)
151 nfv 1874 . . . . . . . . . . . . . . . . . . . . . 22 𝑛(((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)))
152 nfcv 2934 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛𝐴
153 nfcv 2934 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛(𝐹𝑘)
154 nfixp1 8285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑛X𝑛𝐴 (𝐹𝑛)
15510, 154nfcxfr 2932 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛𝑋
156 nfcv 2934 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛(𝑤𝑘)
157155, 156nfmpt 5029 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
158157nfcnv 5603 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛(𝑤𝑋 ↦ (𝑤𝑘))
159 nfcv 2934 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑛𝑢
160158, 159nfima 5783 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑛((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)
161152, 153, 160nfmpo 7060 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
162161nfrn 5672 . . . . . . . . . . . . . . . . . . . . . . 23 𝑛ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
163162nfcri 2928 . . . . . . . . . . . . . . . . . . . . . 22 𝑛 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
164 df-ov 6985 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩)
165119a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V)
166 fveq2 6504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑛 → (𝑤𝑘) = (𝑤𝑛))
167166mpteq2dv 5028 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑛 → (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
168167cnveqd 5600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑛(𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑛)))
169168imaeq1d 5774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑛 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢))
170 imaeq2 5771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = (𝑛) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
171169, 170sylan9eq 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 = 𝑛𝑢 = (𝑛)) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
172 fveq2 6504 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
173171, 172, 133ovmpox 7125 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛) ∧ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ V) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
17429, 35, 165, 173syl3anc 1352 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑛(𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))(𝑛)) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
175164, 174syl5eqr 2830 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
176135ad3antrrr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)): 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝐵)
177176ffnd 6350 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) Fn 𝑘𝐴 ({𝑘} × (𝐹𝑘)))
178 opeliunxp 5473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)) ↔ (𝑛𝐴 ∧ (𝑛) ∈ (𝐹𝑛)))
17929, 35, 178sylanbrc 575 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑛𝐴 ({𝑛} × (𝐹𝑛)))
180 sneq 4454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → {𝑛} = {𝑘})
181 fveq2 6504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
182180, 181xpeq12d 5442 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑘 → ({𝑛} × (𝐹𝑛)) = ({𝑘} × (𝐹𝑘)))
183182cbviunv 4838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑛𝐴 ({𝑛} × (𝐹𝑛)) = 𝑘𝐴 ({𝑘} × (𝐹𝑘))
184179, 183syl6eleq 2878 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ⟨𝑛, (𝑛)⟩ ∈ 𝑘𝐴 ({𝑘} × (𝐹𝑘)))
185 fnfvelrn 6679 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) Fn 𝑘𝐴 ({𝑘} × (𝐹𝑘)) ∧ ⟨𝑛, (𝑛)⟩ ∈ 𝑘𝐴 ({𝑘} × (𝐹𝑘))) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
186177, 184, 185syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))‘⟨𝑛, (𝑛)⟩) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
187175, 186eqeltrrd 2869 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
188 eleq1 2855 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → (𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ↔ ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
189187, 188syl5ibrcom 239 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ 𝑛 ∈ (𝐴𝑚)) → (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
190189ex 405 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑛 ∈ (𝐴𝑚) → (𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
191151, 163, 190rexlimd 3262 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) → 𝑧 ∈ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
192191abssdv 3937 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
193 ssun2 4040 . . . . . . . . . . . . . . . . . . . 20 ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
194192, 193syl6ss 3872 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
195194adantr 473 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
196 simpr 477 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅)
197 simplrl 765 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑚 ∈ Fin)
198 ssfi 8539 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ Fin ∧ (𝐴𝑚) ⊆ 𝑚) → (𝐴𝑚) ∈ Fin)
199197, 84, 198sylancl 578 . . . . . . . . . . . . . . . . . . 19 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝐴𝑚) ∈ Fin)
200 abrexfi 8625 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑚) ∈ Fin → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)
201199, 200syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)
202 elfir 8680 . . . . . . . . . . . . . . . . . 18 ((({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V ∧ ({𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ⊆ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅ ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ Fin)) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
203150, 195, 196, 201, 202syl13anc 1353 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
204120, 203syl5eqel 2872 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
205 elssuni 4746 . . . . . . . . . . . . . . . 16 ( 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
206204, 205syl 17 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
207 fiuni 8693 . . . . . . . . . . . . . . . . . 18 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ∈ V → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
208139, 207syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
209116pwid 4441 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 ∈ 𝒫 𝑋
210209a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211210snssd 4621 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
2121ptuni2 21903 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐵)
21310, 212syl5eq 2828 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 = 𝐵)
214 eqimss2 3916 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑋 = 𝐵 𝐵𝑋)
215213, 214syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵𝑋)
216 sspwuni 4893 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 ⊆ 𝒫 𝑋 𝐵𝑋)
217215, 216sylibr 226 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218136, 217sstrd 3870 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219211, 218unssd 4053 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220 sspwuni 4893 . . . . . . . . . . . . . . . . . . 19 (({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
221219, 220sylib 210 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝑋)
222 elssuni 4746 . . . . . . . . . . . . . . . . . . 19 (𝑋 ∈ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
223145, 222mp1i 13 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋 ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
224221, 223eqssd 3877 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) = 𝑋)
225208, 224eqtr3d 2818 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
226225ad3antrrr 718 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) = 𝑋)
227206, 226sseqtrd 3899 . . . . . . . . . . . . . 14 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)) ⊆ 𝑋)
228227, 52sylib 210 . . . . . . . . . . . . 13 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) = 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛)))
229228, 204eqeltrd 2868 . . . . . . . . . . . 12 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴𝑚)𝑧 = ((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))} ≠ ∅) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
230149, 229pm2.61dane 3057 . . . . . . . . . . 11 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑋 𝑛 ∈ (𝐴𝑚)((𝑤𝑋 ↦ (𝑤𝑛)) “ (𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
231110, 230eqeltrd 2868 . . . . . . . . . 10 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
232231rexlimdvaa 3232 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦))) → (∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
233232impr 447 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2343, 233sylan2b 585 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
235 eleq1 2855 . . . . . . 7 (𝑠 = X𝑦𝐴 (𝑦) → (𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ↔ X𝑦𝐴 (𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
236234, 235syl5ibrcom 239 . . . . . 6 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦))) → (𝑠 = X𝑦𝐴 (𝑦) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
237236expimpd 446 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ((( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
238237exlimdv 1893 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴𝑚)(𝑦) = (𝐹𝑦)) ∧ 𝑠 = X𝑦𝐴 (𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
2392, 238syl5bi 234 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑠𝐵𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))))
240239ssrdv 3866 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
2411ptbasid 21902 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) ∈ 𝐵)
24210, 241syl5eqel 2872 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝑋𝐵)
243242snssd 4621 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244243, 136unssd 4053 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵)
245 fiss 8689 . . . 4 ((𝐵 ∈ TopBases ∧ ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) ⊆ 𝐵) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
246130, 244, 245syl2anc 576 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
2471ptbasin2 21905 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248246, 247sseqtrd 3899 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))) ⊆ 𝐵)
249240, 248eqssd 3877 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  w3a 1069   = wceq 1508  wex 1743  wcel 2051  {cab 2760  wne 2969  wral 3090  wrex 3091  Vcvv 3417  cdif 3828  cun 3829  cin 3830  wss 3831  c0 4181  ifcif 4353  𝒫 cpw 4425  {csn 4444  cop 4450   cuni 4717   cint 4754   ciun 4797   ciin 4798  cmpt 5013   × cxp 5409  ccnv 5410  dom cdm 5411  ran crn 5412  cima 5414   Fn wfn 6188  wf 6189  cfv 6193  (class class class)co 6982  cmpo 6984  Xcixp 8265  Fincfn 8312  ficfi 8675  Topctop 21220  TopBasesctb 21272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-iin 4800  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-1o 7911  df-oadd 7915  df-er 8095  df-ixp 8266  df-en 8313  df-dom 8314  df-fin 8316  df-fi 8676  df-top 21221  df-bases 21273
This theorem is referenced by:  ptval2  21928  xkoptsub  21981  ptcmplem1  22379  prdsxmslem2  22857
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