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Theorem 1stckgenlem 21266
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 795 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2 ssun2 3755 . . . . . . . . 9 {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3 1stckgen.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 1stckgen.3 . . . . . . . . . . 11 (𝜑𝐹(⇝𝑡𝐽)𝐴)
5 lmcl 21011 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝐴) → 𝐴𝑋)
63, 4, 5syl2anc 692 . . . . . . . . . 10 (𝜑𝐴𝑋)
7 snssg 4296 . . . . . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
111, 10sseldd 3584 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 𝑢)
12 eluni2 4406 . . . . . 6 (𝐴 𝑢 ↔ ∃𝑤𝑢 𝐴𝑤)
1311, 12sylib 208 . . . . 5 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑤𝑢 𝐴𝑤)
14 nnuz 11667 . . . . . . 7 ℕ = (ℤ‘1)
15 simprr 795 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐴𝑤)
16 1zzd 11352 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 1 ∈ ℤ)
174ad2antrr 761 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐹(⇝𝑡𝐽)𝐴)
18 simplrl 799 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢 ∈ 𝒫 𝐽)
1918elpwid 4141 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢𝐽)
20 simprl 793 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝑢)
2119, 20sseldd 3584 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝐽)
2214, 15, 16, 17, 21lmcvg 20976 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
23 imassrn 5436 . . . . . . . . . . . . 13 (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24 ssun1 3754 . . . . . . . . . . . . 13 ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
2523, 24sstri 3592 . . . . . . . . . . . 12 (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26 id 22 . . . . . . . . . . . 12 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2725, 26syl5ss 3594 . . . . . . . . . . 11 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ 𝑢)
28 1stckgen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:ℕ⟶𝑋)
29 frn 6010 . . . . . . . . . . . . . . . . . . 19 (𝐹:ℕ⟶𝑋 → ran 𝐹𝑋)
3028, 29syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹𝑋)
3123, 30syl5ss 3594 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
32 resttopon 20875 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
333, 31, 32syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
34 topontop 20641 . . . . . . . . . . . . . . . 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 12712 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑗) ∈ Fin)
37 ffun 6005 . . . . . . . . . . . . . . . . . . . 20 (𝐹:ℕ⟶𝑋 → Fun 𝐹)
3828, 37syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Fun 𝐹)
39 elfznn 12312 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ)
4039ssriv 3587 . . . . . . . . . . . . . . . . . . . 20 (1...𝑗) ⊆ ℕ
41 fdm 6008 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℕ⟶𝑋 → dom 𝐹 = ℕ)
4228, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℕ)
4340, 42syl5sseqr 3633 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑗) ⊆ dom 𝐹)
44 fores 6081 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
4538, 43, 44syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
46 fofi 8196 . . . . . . . . . . . . . . . . . 18 (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
4736, 45, 46syl2anc 692 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
48 pwfi 8205 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
4947, 48sylib 208 . . . . . . . . . . . . . . . 16 (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
50 restsspw 16013 . . . . . . . . . . . . . . . 16 (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
51 ssfi 8124 . . . . . . . . . . . . . . . 16 ((𝒫 (𝐹 “ (1...𝑗)) ∈ Fin ∧ (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
5249, 50, 51sylancl 693 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
5335, 52elind 3776 . . . . . . . . . . . . . 14 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
54 fincmp 21106 . . . . . . . . . . . . . 14 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
5553, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
56 topontop 20641 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
573, 56syl 17 . . . . . . . . . . . . . 14 (𝜑𝐽 ∈ Top)
58 toponuni 20642 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
593, 58syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
6031, 59sseqtrd 3620 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝐽)
61 eqid 2621 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
6261cmpsub 21113 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 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 2927 . . . . . . . . . . 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 648 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
6867adantr 481 . . . . . . . 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 3811 . . . . . . . . . . . . . 14 (𝒫 𝑢 ∩ Fin) ⊆ 𝒫 𝑢
70 simprl 793 . . . . . . . . . . . . . 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 3581 . . . . . . . . . . . . 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 4141 . . . . . . . . . . . 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 801 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → 𝑤𝑢)
7473adantr 481 . . . . . . . . . . . . 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 4309 . . . . . . . . . . . 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 3767 . . . . . . . . . . 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 3189 . . . . . . . . . . . 12 𝑢 ∈ V
7877elpw2 4788 . . . . . . . . . . 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 3812 . . . . . . . . . . . 12 (𝒫 𝑢 ∩ Fin) ⊆ Fin
8180, 70sseldi 3581 . . . . . . . . . . 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 7982 . . . . . . . . . . 11 {𝑤} ∈ Fin
83 unfi 8171 . . . . . . . . . . 11 ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
8481, 82, 83sylancl 693 . . . . . . . . . 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 3776 . . . . . . . . 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 6002 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶𝑋𝐹 Fn ℕ)
8728, 86syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℕ)
8887ad3antrrr 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)) → 𝐹 Fn ℕ)
89 simprrr 804 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
9089adantr 481 . . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
9291eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ 𝑤 ↔ (𝐹𝑛) ∈ 𝑤))
9392rspccva 3294 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ 𝑤)
9490, 93sylan 488 . . . . . . . . . . . . . . . 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 3759 . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . 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 11668 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
9998anbi1i 730 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
100 elfzuzb 12278 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
10199, 100bitr4i 267 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ 𝑛 ∈ (1...𝑗))
102 simprr 795 . . . . . . . . . . . . . . . . . . 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 6204 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
10438, 43, 103syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
105104ad3antrrr 765 . . . . . . . . . . . . . . . . . . 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 2927 . . . . . . . . . . . . . . . . 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 3758 . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . . 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 793 . . . . . . . . . . . . . . . 16 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
113112ad2antlr 762 . . . . . . . . . . . . . . 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 11343 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
115 nnz 11343 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
116 uztric 11653 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
117114, 115, 116syl2an 494 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
118113, 117sylan 488 . . . . . . . . . . . . . 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 826 . . . . . . . . . . . . 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 2960 . . . . . . . . . . . 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 6345 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ ( 𝑠𝑤)) → ran 𝐹 ⊆ ( 𝑠𝑤))
12288, 120, 121syl2anc 692 . . . . . . . . . . 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 3759 . . . . . . . . . . . . . 14 (𝐴𝑤𝐴 ∈ ( 𝑠𝑤))
124123ad2antlr 762 . . . . . . . . . . . . 13 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝐴 ∈ ( 𝑠𝑤))
125124ad2antlr 762 . . . . . . . . . . . 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 4309 . . . . . . . . . . 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 3767 . . . . . . . . . 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 4422 . . . . . . . . . . 11 (𝑠 ∪ {𝑤}) = ( 𝑠 {𝑤})
129 vex 3189 . . . . . . . . . . . . 13 𝑤 ∈ V
130129unisn 4417 . . . . . . . . . . . 12 {𝑤} = 𝑤
131130uneq2i 3742 . . . . . . . . . . 11 ( 𝑠 {𝑤}) = ( 𝑠𝑤)
132128, 131eqtri 2643 . . . . . . . . . 10 (𝑠 ∪ {𝑤}) = ( 𝑠𝑤)
133127, 132syl6sseqr 3631 . . . . . . . . 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 4410 . . . . . . . . . . 11 (𝑣 = (𝑠 ∪ {𝑤}) → 𝑣 = (𝑠 ∪ {𝑤}))
135134sseq2d 3612 . . . . . . . . . 10 (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})))
136135rspcev 3295 . . . . . . . . 9 (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13785, 133, 136syl2anc 692 . . . . . . . 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 3028 . . . . . . 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 679 . . . . . 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 3028 . . . . 5 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
14113, 140rexlimddv 3028 . . . 4 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
142141expr 642 . . 3 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
143142ralrimiva 2960 . 2 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
1446snssd 4309 . . . . 5 (𝜑 → {𝐴} ⊆ 𝑋)
14530, 144unssd 3767 . . . 4 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
146145, 59sseqtrd 3620 . . 3 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝐽)
14761cmpsub 21113 . . 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 692 . 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 383  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402   class class class wbr 4613  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  Fincfn 7899  1c1 9881  cn 10964  cz 11321  cuz 11631  ...cfz 12268  t crest 16002  Topctop 20617  TopOnctopon 20618  𝑡clm 20940  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-lm 20943  df-cmp 21100
This theorem is referenced by:  1stckgen  21267
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