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Theorem 1stcrest 21477
Description: A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
1stcrest ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)

Proof of Theorem 1stcrest
Dummy variables 𝑡 𝑎 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stctop 21467 . . 3 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 resttop 21185 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 561 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 eqid 2771 . . . . . . . 8 𝐽 = 𝐽
54restuni2 21192 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
61, 5sylan 561 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
76eleq2d 2836 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝑥 ∈ (𝐴 𝐽) ↔ 𝑥 (𝐽t 𝐴)))
87biimpar 463 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → 𝑥 ∈ (𝐴 𝐽))
9 simpl 468 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → 𝐽 ∈ 1st𝜔)
10 inss2 3982 . . . . . . 7 (𝐴 𝐽) ⊆ 𝐽
1110sseli 3748 . . . . . 6 (𝑥 ∈ (𝐴 𝐽) → 𝑥 𝐽)
1241stcclb 21468 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
139, 11, 12syl2an 575 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
14 simplll 750 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ 1st𝜔)
15 elpwi 4307 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐽𝑡𝐽)
1615ad2antrl 699 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡𝐽)
17 ssrest 21201 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑡𝐽) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
1814, 16, 17syl2anc 565 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
19 ovex 6823 . . . . . . . 8 (𝐽t 𝐴) ∈ V
2019elpw2 4959 . . . . . . 7 ((𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴) ↔ (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
2118, 20sylibr 224 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴))
22 vex 3354 . . . . . . . 8 𝑡 ∈ V
23 simpllr 752 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐴𝑉)
24 restval 16295 . . . . . . . 8 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
2522, 23, 24sylancr 567 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
26 simprrl 758 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡 ≼ ω)
27 1stcrestlem 21476 . . . . . . . 8 (𝑡 ≼ ω → ran (𝑣𝑡 ↦ (𝑣𝐴)) ≼ ω)
2826, 27syl 17 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → ran (𝑣𝑡 ↦ (𝑣𝐴)) ≼ ω)
2925, 28eqbrtrd 4808 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ≼ ω)
301ad3antrrr 701 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ Top)
31 elrest 16296 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
3230, 23, 31syl2anc 565 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
33 r19.29 3220 . . . . . . . . . . . 12 ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → ∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)))
34 simprr 748 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝑥𝐴)
3534a1d 25 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥𝐴))
3635ancld 532 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦 → (𝑥𝑦𝑥𝐴)))
37 elin 3947 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
3836, 37syl6ibr 242 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥 ∈ (𝑦𝐴)))
39 ssrin 3986 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴))
4039a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴)))
4138, 40anim12d 588 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → ((𝑥𝑦𝑦𝑎) → (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
4241reximdv 3164 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
43 vex 3354 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
4443inex1 4933 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐴) ∈ V
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑦𝑡) → (𝑦𝐴) ∈ V)
46 simp-4r 762 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝐴𝑉)
47 elrest 16296 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
4822, 46, 47sylancr 567 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
49 eleq2 2839 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑥𝑤𝑥 ∈ (𝑦𝐴)))
50 sseq1 3775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑤 ⊆ (𝑎𝐴) ↔ (𝑦𝐴) ⊆ (𝑎𝐴)))
5149, 50anbi12d 608 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (𝑦𝐴) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5251adantl 467 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑤 = (𝑦𝐴)) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5345, 48, 52rexxfr2d 5011 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5442, 53sylibrd 249 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
5554expr 444 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (𝑥𝐴 → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
5655com23 86 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → (𝑥𝐴 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
5756imim2d 57 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (𝑥𝑎 → (𝑥𝐴 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))))
5857imp4b 408 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
59 eleq2 2839 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → (𝑥𝑧𝑥 ∈ (𝑎𝐴)))
60 elin 3947 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑎𝐴) ↔ (𝑥𝑎𝑥𝐴))
6159, 60syl6bb 276 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (𝑥𝑧 ↔ (𝑥𝑎𝑥𝐴)))
62 sseq2 3776 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑎𝐴) → (𝑤𝑧𝑤 ⊆ (𝑎𝐴)))
6362anbi2d 606 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6463rexbidv 3200 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6561, 64imbi12d 333 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐴) → ((𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)) ↔ ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
6658, 65syl5ibrcom 237 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → (𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6766expimpd 441 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6867rexlimdva 3179 . . . . . . . . . . . 12 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6933, 68syl5 34 . . . . . . . . . . 11 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7069expd 400 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
7170impr 442 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7271adantrrl 695 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7332, 72sylbid 230 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑧 ∈ (𝐽t 𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7473ralrimiv 3114 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
75 breq1 4789 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (𝑦 ≼ ω ↔ (𝑡t 𝐴) ≼ ω))
76 rexeq 3288 . . . . . . . . . 10 (𝑦 = (𝑡t 𝐴) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
7776imbi2d 329 . . . . . . . . 9 (𝑦 = (𝑡t 𝐴) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7877ralbidv 3135 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7975, 78anbi12d 608 . . . . . . 7 (𝑦 = (𝑡t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
8079rspcev 3460 . . . . . 6 (((𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴) ∧ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8121, 29, 74, 80syl12anc 1474 . . . . 5 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8213, 81rexlimddv 3183 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
838, 82syldan 571 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8483ralrimiva 3115 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
85 eqid 2771 . . 3 (𝐽t 𝐴) = (𝐽t 𝐴)
8685is1stc2 21466 . 2 ((𝐽t 𝐴) ∈ 1st𝜔 ↔ ((𝐽t 𝐴) ∈ Top ∧ ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
873, 84, 86sylanbrc 564 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574   class class class wbr 4786  cmpt 4863  ran crn 5250  (class class class)co 6793  ωcom 7212  cdom 8107  t crest 16289  Topctop 20918  1st𝜔c1stc 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-fin 8113  df-fi 8473  df-card 8965  df-acn 8968  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-1stc 21463
This theorem is referenced by:  lly1stc  21520
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