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Theorem 1stckgenlem 23561
Description: The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
1stckgen.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
1stckgen.2 (𝜑𝐹:ℕ⟶𝑋)
1stckgen.3 (𝜑𝐹(⇝𝑡𝐽)𝐴)
Assertion
Ref Expression
1stckgenlem (𝜑 → (𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp)

Proof of Theorem 1stckgenlem
Dummy variables 𝑗 𝑘 𝑛 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2 ssun2 4179 . . . . . . . . 9 {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3 1stckgen.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 1stckgen.3 . . . . . . . . . . 11 (𝜑𝐹(⇝𝑡𝐽)𝐴)
5 lmcl 23305 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝐴) → 𝐴𝑋)
63, 4, 5syl2anc 584 . . . . . . . . . 10 (𝜑𝐴𝑋)
7 snssg 4783 . . . . . . . . . 10 (𝐴𝑋 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∈ (ran 𝐹 ∪ {𝐴}) ↔ {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})))
92, 8mpbiri 258 . . . . . . . 8 (𝜑𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
109adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
111, 10sseldd 3984 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 𝑢)
12 eluni2 4911 . . . . . 6 (𝐴 𝑢 ↔ ∃𝑤𝑢 𝐴𝑤)
1311, 12sylib 218 . . . . 5 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑤𝑢 𝐴𝑤)
14 nnuz 12921 . . . . . . 7 ℕ = (ℤ‘1)
15 simprr 773 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐴𝑤)
16 1zzd 12648 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 1 ∈ ℤ)
174ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐹(⇝𝑡𝐽)𝐴)
18 simplrl 777 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢 ∈ 𝒫 𝐽)
1918elpwid 4609 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢𝐽)
20 simprl 771 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝑢)
2119, 20sseldd 3984 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝐽)
2214, 15, 16, 17, 21lmcvg 23270 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
23 imassrn 6089 . . . . . . . . . . . . 13 (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24 ssun1 4178 . . . . . . . . . . . . 13 ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
2523, 24sstri 3993 . . . . . . . . . . . 12 (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26 id 22 . . . . . . . . . . . 12 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2725, 26sstrid 3995 . . . . . . . . . . 11 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ 𝑢)
28 1stckgen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:ℕ⟶𝑋)
2928frnd 6744 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹𝑋)
3023, 29sstrid 3995 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31 resttopon 23169 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
323, 30, 31syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
33 topontop 22919 . . . . . . . . . . . . . . . 16 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Top)
3432, 33syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Top)
35 fzfid 14014 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑗) ∈ Fin)
3628ffund 6740 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Fun 𝐹)
37 fz1ssnn 13595 . . . . . . . . . . . . . . . . . . . 20 (1...𝑗) ⊆ ℕ
3828fdmd 6746 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℕ)
3937, 38sseqtrrid 4027 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40 fores 6830 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
4136, 39, 40syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
42 fofi 9351 . . . . . . . . . . . . . . . . . 18 (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
4335, 41, 42syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
44 pwfi 9357 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
4543, 44sylib 218 . . . . . . . . . . . . . . . 16 (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46 restsspw 17476 . . . . . . . . . . . . . . . 16 (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47 ssfi 9213 . . . . . . . . . . . . . . . 16 ((𝒫 (𝐹 “ (1...𝑗)) ∈ Fin ∧ (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
4845, 46, 47sylancl 586 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
4934, 48elind 4200 . . . . . . . . . . . . . 14 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50 fincmp 23401 . . . . . . . . . . . . . 14 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
5149, 50syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
52 topontop 22919 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
533, 52syl 17 . . . . . . . . . . . . . 14 (𝜑𝐽 ∈ Top)
54 toponuni 22920 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
553, 54syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
5630, 55sseqtrd 4020 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝐽)
57 eqid 2737 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
5857cmpsub 23408 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝐹 “ (1...𝑗)) ⊆ 𝐽) → ((𝐽t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)))
5953, 56, 58syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐽t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)))
6051, 59mpbid 232 . . . . . . . . . . . 12 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6160r19.21bi 3251 . . . . . . . . . . 11 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6227, 61syl5 34 . . . . . . . . . 10 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6362impr 454 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
6463adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
65 simprl 771 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑠 ∈ (𝒫 𝑢 ∩ Fin))
6665elin1d 4204 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑠 ∈ 𝒫 𝑢)
6766elpwid 4609 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑠𝑢)
68 simprll 779 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → 𝑤𝑢)
6968adantr 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑤𝑢)
7069snssd 4809 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → {𝑤} ⊆ 𝑢)
7167, 70unssd 4192 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (𝑠 ∪ {𝑤}) ⊆ 𝑢)
72 vex 3484 . . . . . . . . . . . 12 𝑢 ∈ V
7372elpw2 5334 . . . . . . . . . . 11 ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑢)
7471, 73sylibr 234 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢)
7565elin2d 4205 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑠 ∈ Fin)
76 snfi 9083 . . . . . . . . . . 11 {𝑤} ∈ Fin
77 unfi 9211 . . . . . . . . . . 11 ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
7875, 76, 77sylancl 586 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ Fin)
7974, 78elind 4200 . . . . . . . . 9 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin))
8028ffnd 6737 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℕ)
8180ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝐹 Fn ℕ)
82 simprrr 782 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
8382adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
84 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8584eleq1d 2826 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ 𝑤 ↔ (𝐹𝑛) ∈ 𝑤))
8685rspccva 3621 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ 𝑤)
8783, 86sylan 580 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ 𝑤)
88 elun2 4183 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ 𝑤 → (𝐹𝑛) ∈ ( 𝑠𝑤))
8987, 88syl 17 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ ( 𝑠𝑤))
9089adantlr 715 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ ( 𝑠𝑤))
91 elnnuz 12922 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
9291anbi1i 624 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
93 elfzuzb 13558 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
9492, 93bitr4i 278 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ 𝑛 ∈ (1...𝑗))
95 simprr 773 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (𝐹 “ (1...𝑗)) ⊆ 𝑠)
96 funimass4 6973 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
9736, 39, 96syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
9897ad3antrrr 730 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
9995, 98mpbid 232 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠)
10099r19.21bi 3251 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ (1...𝑗)) → (𝐹𝑛) ∈ 𝑠)
101 elun1 4182 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ 𝑠 → (𝐹𝑛) ∈ ( 𝑠𝑤))
102100, 101syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ (1...𝑗)) → (𝐹𝑛) ∈ ( 𝑠𝑤))
10394, 102sylan2b 594 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ (𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛))) → (𝐹𝑛) ∈ ( 𝑠𝑤))
104103anassrs 467 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ ( 𝑠𝑤))
105 simprl 771 . . . . . . . . . . . . . . . 16 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
106105ad2antlr 727 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝑗 ∈ ℕ)
107 nnz 12634 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108 nnz 12634 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109 uztric 12902 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
110107, 108, 109syl2an 596 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
111106, 110sylan 580 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
11290, 104, 111mpjaodan 961 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( 𝑠𝑤))
113112ralrimiva 3146 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ ( 𝑠𝑤))
114 fnfvrnss 7141 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ ( 𝑠𝑤)) → ran 𝐹 ⊆ ( 𝑠𝑤))
11581, 113, 114syl2anc 584 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ran 𝐹 ⊆ ( 𝑠𝑤))
116 elun2 4183 . . . . . . . . . . . . . 14 (𝐴𝑤𝐴 ∈ ( 𝑠𝑤))
117116ad2antlr 727 . . . . . . . . . . . . 13 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝐴 ∈ ( 𝑠𝑤))
118117ad2antlr 727 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → 𝐴 ∈ ( 𝑠𝑤))
119118snssd 4809 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → {𝐴} ⊆ ( 𝑠𝑤))
120115, 119unssd 4192 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (ran 𝐹 ∪ {𝐴}) ⊆ ( 𝑠𝑤))
121 uniun 4930 . . . . . . . . . . 11 (𝑠 ∪ {𝑤}) = ( 𝑠 {𝑤})
122 unisnv 4927 . . . . . . . . . . . 12 {𝑤} = 𝑤
123122uneq2i 4165 . . . . . . . . . . 11 ( 𝑠 {𝑤}) = ( 𝑠𝑤)
124121, 123eqtri 2765 . . . . . . . . . 10 (𝑠 ∪ {𝑤}) = ( 𝑠𝑤)
125120, 124sseqtrrdi 4025 . . . . . . . . 9 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤}))
126 unieq 4918 . . . . . . . . . . 11 (𝑣 = (𝑠 ∪ {𝑤}) → 𝑣 = (𝑠 ∪ {𝑤}))
127126sseq2d 4016 . . . . . . . . . 10 (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})))
128127rspcev 3622 . . . . . . . . 9 (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
12979, 125, 128syl2anc 584 . . . . . . . 8 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) ∧ (𝑠 ∈ (𝒫 𝑢 ∩ Fin) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13064, 129rexlimddv 3161 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
131130anassrs 467 . . . . . 6 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13222, 131rexlimddv 3161 . . . . 5 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13313, 132rexlimddv 3161 . . . 4 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
134133expr 456 . . 3 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
135134ralrimiva 3146 . 2 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
1366snssd 4809 . . . . 5 (𝜑 → {𝐴} ⊆ 𝑋)
13729, 136unssd 4192 . . . 4 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
138137, 55sseqtrd 4020 . . 3 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝐽)
13957cmpsub 23408 . . 3 ((𝐽 ∈ Top ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝐽) → ((𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)))
14053, 138, 139syl2anc 584 . 2 (𝜑 → ((𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)))
141135, 140mpbird 257 1 (𝜑 → (𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  Fincfn 8985  1c1 11156  cn 12266  cz 12613  cuz 12878  ...cfz 13547  t crest 17465  Topctop 22899  TopOnctopon 22916  𝑡clm 23234  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-lm 23237  df-cmp 23395
This theorem is referenced by:  1stckgen  23562
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