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Theorem 1stckgenlem 22168
 Description: The one-point compactification of ℕ 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 772 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2 ssun2 4100 . . . . . . . . 9 {𝐴} ⊆ (ran 𝐹 ∪ {𝐴})
3 1stckgen.1 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 1stckgen.3 . . . . . . . . . . 11 (𝜑𝐹(⇝𝑡𝐽)𝐴)
5 lmcl 21912 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝐴) → 𝐴𝑋)
63, 4, 5syl2anc 587 . . . . . . . . . 10 (𝜑𝐴𝑋)
7 snssg 4678 . . . . . . . . . 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 261 . . . . . . . 8 (𝜑𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
109adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 ∈ (ran 𝐹 ∪ {𝐴}))
111, 10sseldd 3916 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → 𝐴 𝑢)
12 eluni2 4805 . . . . . 6 (𝐴 𝑢 ↔ ∃𝑤𝑢 𝐴𝑤)
1311, 12sylib 221 . . . . 5 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑤𝑢 𝐴𝑤)
14 nnuz 12272 . . . . . . 7 ℕ = (ℤ‘1)
15 simprr 772 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐴𝑤)
16 1zzd 12004 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 1 ∈ ℤ)
174ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝐹(⇝𝑡𝐽)𝐴)
18 simplrl 776 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢 ∈ 𝒫 𝐽)
1918elpwid 4508 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑢𝐽)
20 simprl 770 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝑢)
2119, 20sseldd 3916 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → 𝑤𝐽)
2214, 15, 16, 17, 21lmcvg 21877 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
23 imassrn 5908 . . . . . . . . . . . . 13 (𝐹 “ (1...𝑗)) ⊆ ran 𝐹
24 ssun1 4099 . . . . . . . . . . . . 13 ran 𝐹 ⊆ (ran 𝐹 ∪ {𝐴})
2523, 24sstri 3924 . . . . . . . . . . . 12 (𝐹 “ (1...𝑗)) ⊆ (ran 𝐹 ∪ {𝐴})
26 id 22 . . . . . . . . . . . 12 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)
2725, 26sstrid 3926 . . . . . . . . . . 11 ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → (𝐹 “ (1...𝑗)) ⊆ 𝑢)
28 1stckgen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:ℕ⟶𝑋)
2928frnd 6495 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹𝑋)
3023, 29sstrid 3926 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝑋)
31 resttopon 21776 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹 “ (1...𝑗)) ⊆ 𝑋) → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
323, 30, 31syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (TopOn‘(𝐹 “ (1...𝑗))))
33 topontop 21528 . . . . . . . . . . . . . . . 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 13339 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑗) ∈ Fin)
3628ffund 6492 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Fun 𝐹)
37 fz1ssnn 12936 . . . . . . . . . . . . . . . . . . . 20 (1...𝑗) ⊆ ℕ
3828fdmd 6498 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℕ)
3937, 38sseqtrrid 3968 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑗) ⊆ dom 𝐹)
40 fores 6576 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
4136, 39, 40syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗)))
42 fofi 8797 . . . . . . . . . . . . . . . . . 18 (((1...𝑗) ∈ Fin ∧ (𝐹 ↾ (1...𝑗)):(1...𝑗)–onto→(𝐹 “ (1...𝑗))) → (𝐹 “ (1...𝑗)) ∈ Fin)
4335, 41, 42syl2anc 587 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (1...𝑗)) ∈ Fin)
44 pwfi 8806 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (1...𝑗)) ∈ Fin ↔ 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
4543, 44sylib 221 . . . . . . . . . . . . . . . 16 (𝜑 → 𝒫 (𝐹 “ (1...𝑗)) ∈ Fin)
46 restsspw 16700 . . . . . . . . . . . . . . . 16 (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))
47 ssfi 8725 . . . . . . . . . . . . . . . 16 ((𝒫 (𝐹 “ (1...𝑗)) ∈ Fin ∧ (𝐽t (𝐹 “ (1...𝑗))) ⊆ 𝒫 (𝐹 “ (1...𝑗))) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
4845, 46, 47sylancl 589 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Fin)
4934, 48elind 4121 . . . . . . . . . . . . . 14 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin))
50 fincmp 22008 . . . . . . . . . . . . . 14 ((𝐽t (𝐹 “ (1...𝑗))) ∈ (Top ∩ Fin) → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
5149, 50syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐽t (𝐹 “ (1...𝑗))) ∈ Comp)
52 topontop 21528 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
533, 52syl 17 . . . . . . . . . . . . . 14 (𝜑𝐽 ∈ Top)
54 toponuni 21529 . . . . . . . . . . . . . . . 16 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
553, 54syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
5630, 55sseqtrd 3955 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (1...𝑗)) ⊆ 𝐽)
57 eqid 2798 . . . . . . . . . . . . . . 15 𝐽 = 𝐽
5857cmpsub 22015 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 13 (𝜑 → ((𝐽t (𝐹 “ (1...𝑗))) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)))
6051, 59mpbid 235 . . . . . . . . . . . 12 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((𝐹 “ (1...𝑗)) ⊆ 𝑢 → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠))
6160r19.21bi 3173 . . . . . . . . . . 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 458 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑠 ∈ (𝒫 𝑢 ∩ Fin)(𝐹 “ (1...𝑗)) ⊆ 𝑠)
6463adantr 484 . . . . . . . 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 770 . . . . . . . . . . . . . 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 4125 . . . . . . . . . . . . 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 4508 . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → 𝑤𝑢)
6968adantr 484 . . . . . . . . . . . . 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 4702 . . . . . . . . . . . 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 4113 . . . . . . . . . . 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 3444 . . . . . . . . . . . 12 𝑢 ∈ V
7372elpw2 5213 . . . . . . . . . . 11 ((𝑠 ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (𝑠 ∪ {𝑤}) ⊆ 𝑢)
7471, 73sylibr 237 . . . . . . . . . 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 4126 . . . . . . . . . . 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 8580 . . . . . . . . . . 11 {𝑤} ∈ Fin
77 unfi 8772 . . . . . . . . . . 11 ((𝑠 ∈ Fin ∧ {𝑤} ∈ Fin) → (𝑠 ∪ {𝑤}) ∈ Fin)
7875, 76, 77sylancl 589 . . . . . . . . . 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 4121 . . . . . . . . 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 6489 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℕ)
8180ad3antrrr 729 . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)
8382adantr 484 . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8584eleq1d 2874 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ 𝑤 ↔ (𝐹𝑛) ∈ 𝑤))
8685rspccva 3570 . . . . . . . . . . . . . . . . 17 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤𝑛 ∈ (ℤ𝑗)) → (𝐹𝑛) ∈ 𝑤)
8783, 86sylan 583 . . . . . . . . . . . . . . . 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 4104 . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . 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 12273 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
9291anbi1i 626 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
93 elfzuzb 12899 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑗) ↔ (𝑛 ∈ (ℤ‘1) ∧ 𝑗 ∈ (ℤ𝑛)))
9492, 93bitr4i 281 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑛)) ↔ 𝑛 ∈ (1...𝑗))
95 simprr 772 . . . . . . . . . . . . . . . . . . 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 6706 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (1...𝑗) ⊆ dom 𝐹) → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
9736, 39, 96syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹 “ (1...𝑗)) ⊆ 𝑠 ↔ ∀𝑛 ∈ (1...𝑗)(𝐹𝑛) ∈ 𝑠))
9897ad3antrrr 729 . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . . . . . 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 4103 . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 16 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝑗 ∈ ℕ)
106105ad2antlr 726 . . . . . . . . . . . . . . 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 11995 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
108 nnz 11995 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
109 uztric 12257 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
110107, 108, 109syl2an 598 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (ℤ𝑗) ∨ 𝑗 ∈ (ℤ𝑛)))
111106, 110sylan 583 . . . . . . . . . . . . . 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 956 . . . . . . . . . . . . 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 3149 . . . . . . . . . . . 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 6862 . . . . . . . . . . . 12 ((𝐹 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ ( 𝑠𝑤)) → ran 𝐹 ⊆ ( 𝑠𝑤))
11581, 113, 114syl2anc 587 . . . . . . . . . . 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 4104 . . . . . . . . . . . . . 14 (𝐴𝑤𝐴 ∈ ( 𝑠𝑤))
117116ad2antlr 726 . . . . . . . . . . . . 13 (((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → 𝐴 ∈ ( 𝑠𝑤))
118117ad2antlr 726 . . . . . . . . . . . 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 4702 . . . . . . . . . . 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 4113 . . . . . . . . . 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 4824 . . . . . . . . . . 11 (𝑠 ∪ {𝑤}) = ( 𝑠 {𝑤})
122 vex 3444 . . . . . . . . . . . . 13 𝑤 ∈ V
123122unisn 4821 . . . . . . . . . . . 12 {𝑤} = 𝑤
124123uneq2i 4087 . . . . . . . . . . 11 ( 𝑠 {𝑤}) = ( 𝑠𝑤)
125121, 124eqtri 2821 . . . . . . . . . 10 (𝑠 ∪ {𝑤}) = ( 𝑠𝑤)
126120, 125sseqtrrdi 3966 . . . . . . . . 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}))
127 unieq 4812 . . . . . . . . . . 11 (𝑣 = (𝑠 ∪ {𝑤}) → 𝑣 = (𝑠 ∪ {𝑤}))
128127sseq2d 3947 . . . . . . . . . 10 (𝑣 = (𝑠 ∪ {𝑤}) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑣 ↔ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})))
129128rspcev 3571 . . . . . . . . 9 (((𝑠 ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ (ran 𝐹 ∪ {𝐴}) ⊆ (𝑠 ∪ {𝑤})) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13079, 126, 129syl2anc 587 . . . . . . . 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)
13164, 130rexlimddv 3250 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ ((𝑤𝑢𝐴𝑤) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
132131anassrs 471 . . . . . 6 ((((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) ∧ (𝑗 ∈ ℕ ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13322, 132rexlimddv 3250 . . . . 5 (((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) ∧ (𝑤𝑢𝐴𝑤)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
13413, 133rexlimddv 3250 . . . 4 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐽 ∧ (ran 𝐹 ∪ {𝐴}) ⊆ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)
135134expr 460 . . 3 ((𝜑𝑢 ∈ 𝒫 𝐽) → ((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
136135ralrimiva 3149 . 2 (𝜑 → ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣))
1376snssd 4702 . . . . 5 (𝜑 → {𝐴} ⊆ 𝑋)
13829, 137unssd 4113 . . . 4 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝑋)
139138, 55sseqtrd 3955 . . 3 (𝜑 → (ran 𝐹 ∪ {𝐴}) ⊆ 𝐽)
14057cmpsub 22015 . . 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)))
14153, 139, 140syl2anc 587 . 2 (𝜑 → ((𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐽((ran 𝐹 ∪ {𝐴}) ⊆ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(ran 𝐹 ∪ {𝐴}) ⊆ 𝑣)))
142136, 141mpbird 260 1 (𝜑 → (𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   class class class wbr 5031  dom cdm 5520  ran crn 5521   ↾ cres 5522   “ cima 5523  Fun wfun 6319   Fn wfn 6320  ⟶wf 6321  –onto→wfo 6323  ‘cfv 6325  (class class class)co 7136  Fincfn 8495  1c1 10530  ℕcn 11628  ℤcz 11972  ℤ≥cuz 12234  ...cfz 12888   ↾t crest 16689  Topctop 21508  TopOnctopon 21525  ⇝𝑡clm 21841  Compccmp 22001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fi 8862  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-rest 16691  df-topgen 16712  df-top 21509  df-topon 21526  df-bases 21561  df-lm 21844  df-cmp 22002 This theorem is referenced by:  1stckgen  22169
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