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Theorem 1stckgenlem 21558
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 813 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2 ssun2 3920 . . . . . . . . 9 {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3 1stckgen.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 1stckgen.3 . . . . . . . . . . 11 (𝜑𝐹(⇝𝑡𝐽)𝐴)
5 lmcl 21303 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝐴) → 𝐴𝑋)
63, 4, 5syl2anc 696 . . . . . . . . . 10 (𝜑𝐴𝑋)
7 snssg 4459 . . . . . . . . . 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 248 . . . . . . . 8 (𝜑𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
109adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
111, 10sseldd 3745 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 𝑢)
12 eluni2 4592 . . . . . 6 (𝐴 𝑢 ↔ ∃𝑤𝑢 𝐴𝑤)
1311, 12sylib 208 . . . . 5 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑤𝑢 𝐴𝑤)
14 nnuz 11916 . . . . . . 7 ℕ = (ℤ‘1)
15 simprr 813 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐴𝑤)
16 1zzd 11600 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 1 ∈ ℤ)
174ad2antrr 764 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐹(⇝𝑡𝐽)𝐴)
18 simplrl 819 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢 ∈ 𝒫 𝐽)
1918elpwid 4314 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢𝐽)
20 simprl 811 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝑢)
2119, 20sseldd 3745 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝐽)
2214, 15, 16, 17, 21lmcvg 21268 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
23 imassrn 5635 . . . . . . . . . . . . 13 (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24 ssun1 3919 . . . . . . . . . . . . 13 ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
2523, 24sstri 3753 . . . . . . . . . . . 12 (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26 id 22 . . . . . . . . . . . 12 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2725, 26syl5ss 3755 . . . . . . . . . . 11 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ 𝑢)
28 1stckgen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:ℕ⟶𝑋)
29 frn 6214 . . . . . . . . . . . . . . . . . . 19 (𝐹:ℕ⟶𝑋 → ran 𝐹𝑋)
3028, 29syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹𝑋)
3123, 30syl5ss 3755 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
32 resttopon 21167 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
333, 31, 32syl2anc 696 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
34 topontop 20920 . . . . . . . . . . . . . . . 16 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Top)
3533, 34syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Top)
36 fzfid 12966 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑗) ∈ Fin)
37 ffun 6209 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ℕ⟶𝑋 → Fun 𝐹)
3828, 37syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Fun 𝐹)
39 elfznn 12563 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ)
4039ssriv 3748 . . . . . . . . . . . . . . . . . . . 20 (1...𝑗) ⊆ ℕ
41 fdm 6212 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℕ⟶𝑋 → dom 𝐹 = ℕ)
4228, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℕ)
4340, 42syl5sseqr 3795 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑗) ⊆ dom 𝐹)
44 fores 6285 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
4538, 43, 44syl2anc 696 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
46 fofi 8417 . . . . . . . . . . . . . . . . . 18 (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
4736, 45, 46syl2anc 696 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
48 pwfi 8426 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
4947, 48sylib 208 . . . . . . . . . . . . . . . 16 (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
50 restsspw 16294 . . . . . . . . . . . . . . . 16 (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
51 ssfi 8345 . . . . . . . . . . . . . . . 16 ((𝒫 (𝐹 “ (1...𝑗)) ∈ Fin ∧ (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
5249, 50, 51sylancl 697 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
5335, 52elind 3941 . . . . . . . . . . . . . 14 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
54 fincmp 21398 . . . . . . . . . . . . . 14 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
5553, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
56 topontop 20920 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
573, 56syl 17 . . . . . . . . . . . . . 14 (𝜑𝐽 ∈ Top)
58 toponuni 20921 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
593, 58syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
6031, 59sseqtrd 3782 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝐽)
61 eqid 2760 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
6261cmpsub 21405 . . . . . . . . . . . . . 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)))
6357, 60, 62syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → ((𝐽t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)))
6455, 63mpbid 222 . . . . . . . . . . . 12 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6564r19.21bi 3070 . . . . . . . . . . 11 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6627, 65syl5 34 . . . . . . . . . 10 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6766impr 650 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
6867adantr 472 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
69 inss1 3976 . . . . . . . . . . . . . 14 (𝒫 𝑢 ∩ Fin) ⊆ 𝒫 𝑢
70 simprl 811 . . . . . . . . . . . . . 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))
7169, 70sseldi 3742 . . . . . . . . . . . . 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)
7271elpwid 4314 . . . . . . . . . . . 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)
73 simprll 821 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → 𝑤𝑢)
7473adantr 472 . . . . . . . . . . . . 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)
7574snssd 4485 . . . . . . . . . . . 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)
7672, 75unssd 3932 . . . . . . . . . . 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)
77 vex 3343 . . . . . . . . . . . 12 𝑢 ∈ V
7877elpw2 4977 . . . . . . . . . . 11 ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑢)
7976, 78sylibr 224 . . . . . . . . . 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)
80 inss2 3977 . . . . . . . . . . . 12 (𝒫 𝑢 ∩ Fin) ⊆ Fin
8180, 70sseldi 3742 . . . . . . . . . . 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)
82 snfi 8203 . . . . . . . . . . 11 {𝑤} ∈ Fin
83 unfi 8392 . . . . . . . . . . 11 ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
8481, 82, 83sylancl 697 . . . . . . . . . 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)
8579, 84elind 3941 . . . . . . . . 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))
86 ffn 6206 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶𝑋𝐹 Fn ℕ)
8728, 86syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℕ)
8887ad3antrrr 768 . . . . . . . . . . . 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)
89 simprrr 824 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
9089adantr 472 . . . . . . . . . . . . . . . . 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)
91 fveq2 6352 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
9291eleq1d 2824 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ 𝑤 ↔ (𝐹𝑛) ∈ 𝑤))
9392rspccva 3448 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ 𝑤)
9490, 93sylan 489 . . . . . . . . . . . . . . . 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)
95 elun2 3924 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ 𝑤 → (𝐹𝑛) ∈ ( 𝑠𝑤))
9694, 95syl 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))
9796adantlr 753 . . . . . . . . . . . . . 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))
98 elnnuz 11917 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
9998anbi1i 733 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
100 elfzuzb 12529 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
10199, 100bitr4i 267 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ 𝑛 ∈ (1...𝑗))
102 simprr 813 . . . . . . . . . . . . . . . . . . 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)
103 funimass4 6409 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
10438, 43, 103syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
105104ad3antrrr 768 . . . . . . . . . . . . . . . . . . 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))
106102, 105mpbid 222 . . . . . . . . . . . . . . . . . 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)
107106r19.21bi 3070 . . . . . . . . . . . . . . . . 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)
108 elun1 3923 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ 𝑠 → (𝐹𝑛) ∈ ( 𝑠𝑤))
109107, 108syl 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))
110101, 109sylan2b 493 . . . . . . . . . . . . . . 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))
111110anassrs 683 . . . . . . . . . . . . . 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))
112 simprl 811 . . . . . . . . . . . . . . . 16 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
113112ad2antlr 765 . . . . . . . . . . . . . . 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)
114 nnz 11591 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
115 nnz 11591 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
116 uztric 11901 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
117114, 115, 116syl2an 495 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
118113, 117sylan 489 . . . . . . . . . . . . . 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)))
11997, 111, 118mpjaodan 862 . . . . . . . . . . . . 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))
120119ralrimiva 3104 . . . . . . . . . . . 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))
121 fnfvrnss 6553 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ ( 𝑠𝑤)) → ran 𝐹 ⊆ ( 𝑠𝑤))
12288, 120, 121syl2anc 696 . . . . . . . . . . 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))
123 elun2 3924 . . . . . . . . . . . . . 14 (𝐴𝑤𝐴 ∈ ( 𝑠𝑤))
124123ad2antlr 765 . . . . . . . . . . . . 13 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝐴 ∈ ( 𝑠𝑤))
125124ad2antlr 765 . . . . . . . . . . . 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))
126125snssd 4485 . . . . . . . . . . 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))
127122, 126unssd 3932 . . . . . . . . . 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))
128 uniun 4608 . . . . . . . . . . 11 (𝑠 ∪ {𝑤}) = ( 𝑠 {𝑤})
129 vex 3343 . . . . . . . . . . . . 13 𝑤 ∈ V
130129unisn 4603 . . . . . . . . . . . 12 {𝑤} = 𝑤
131130uneq2i 3907 . . . . . . . . . . 11 ( 𝑠 {𝑤}) = ( 𝑠𝑤)
132128, 131eqtri 2782 . . . . . . . . . 10 (𝑠 ∪ {𝑤}) = ( 𝑠𝑤)
133127, 132syl6sseqr 3793 . . . . . . . . 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}))
134 unieq 4596 . . . . . . . . . . 11 (𝑣 = (𝑠 ∪ {𝑤}) → 𝑣 = (𝑠 ∪ {𝑤}))
135134sseq2d 3774 . . . . . . . . . 10 (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})))
136135rspcev 3449 . . . . . . . . 9 (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13785, 133, 136syl2anc 696 . . . . . . . 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)
13868, 137rexlimddv 3173 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
139138anassrs 683 . . . . . 6 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
14022, 139rexlimddv 3173 . . . . 5 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
14113, 140rexlimddv 3173 . . . 4 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
142141expr 644 . . 3 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
143142ralrimiva 3104 . 2 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
1446snssd 4485 . . . . 5 (𝜑 → {𝐴} ⊆ 𝑋)
14530, 144unssd 3932 . . . 4 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
146145, 59sseqtrd 3782 . . 3 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝐽)
14761cmpsub 21405 . . 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)))
14857, 146, 147syl2anc 696 . 2 (𝜑 → ((𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)))
149143, 148mpbird 247 1 (𝜑 → (𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cun 3713  cin 3714  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Fun wfun 6043   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  Fincfn 8121  1c1 10129  cn 11212  cz 11569  cuz 11879  ...cfz 12519  t crest 16283  Topctop 20900  TopOnctopon 20917  𝑡clm 21232  Compccmp 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-lm 21235  df-cmp 21392
This theorem is referenced by:  1stckgen  21559
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