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Theorem 1stcrest 21464
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 21454 . . 3 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 resttop 21172 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 571 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 eqid 2802 . . . . . . . 8 𝐽 = 𝐽
54restuni2 21179 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
61, 5sylan 571 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
76eleq2d 2867 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝑥 ∈ (𝐴 𝐽) ↔ 𝑥 (𝐽t 𝐴)))
87biimpar 465 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → 𝑥 ∈ (𝐴 𝐽))
9 simpl 470 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → 𝐽 ∈ 1st𝜔)
10 inss2 4024 . . . . . . 7 (𝐴 𝐽) ⊆ 𝐽
1110sseli 3788 . . . . . 6 (𝑥 ∈ (𝐴 𝐽) → 𝑥 𝐽)
1241stcclb 21455 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
139, 11, 12syl2an 585 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
14 simplll 782 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ 1st𝜔)
15 elpwi 4355 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐽𝑡𝐽)
1615ad2antrl 710 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡𝐽)
17 ssrest 21188 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑡𝐽) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
1814, 16, 17syl2anc 575 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
19 ovex 6900 . . . . . . . 8 (𝐽t 𝐴) ∈ V
2019elpw2 5014 . . . . . . 7 ((𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴) ↔ (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
2118, 20sylibr 225 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴))
22 vex 3390 . . . . . . . 8 𝑡 ∈ V
23 simpllr 784 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐴𝑉)
24 restval 16286 . . . . . . . 8 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
2522, 23, 24sylancr 577 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
26 simprrl 790 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡 ≼ ω)
27 1stcrestlem 21463 . . . . . . . 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 4859 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ≼ ω)
301ad3antrrr 712 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ Top)
31 elrest 16287 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
3230, 23, 31syl2anc 575 . . . . . . . 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 3256 . . . . . . . . . . . 12 ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → ∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)))
34 simprr 780 . . . . . . . . . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦 → (𝑥𝑦𝑥𝐴)))
37 elin 3989 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
3836, 37syl6ibr 243 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥 ∈ (𝑦𝐴)))
39 ssrin 4028 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴))
4039a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴)))
4138, 40anim12d 598 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → ((𝑥𝑦𝑦𝑎) → (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
4241reximdv 3199 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
43 vex 3390 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
4443inex1 4988 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐴) ∈ V
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑦𝑡) → (𝑦𝐴) ∈ V)
46 simp-4r 794 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝐴𝑉)
47 elrest 16287 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
4822, 46, 47sylancr 577 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
49 eleq2 2870 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑥𝑤𝑥 ∈ (𝑦𝐴)))
50 sseq1 3817 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑤 ⊆ (𝑎𝐴) ↔ (𝑦𝐴) ⊆ (𝑎𝐴)))
5149, 50anbi12d 618 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (𝑦𝐴) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5251adantl 469 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑤 = (𝑦𝐴)) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5345, 48, 52rexxfr2d 5074 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5442, 53sylibrd 250 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
5554expr 446 . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
59 eleq2 2870 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → (𝑥𝑧𝑥 ∈ (𝑎𝐴)))
60 elin 3989 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑎𝐴) ↔ (𝑥𝑎𝑥𝐴))
6159, 60syl6bb 278 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (𝑥𝑧 ↔ (𝑥𝑎𝑥𝐴)))
62 sseq2 3818 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑎𝐴) → (𝑤𝑧𝑤 ⊆ (𝑎𝐴)))
6362anbi2d 616 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6463rexbidv 3236 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6561, 64imbi12d 335 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐴) → ((𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)) ↔ ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
6658, 65syl5ibrcom 238 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → (𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6766expimpd 443 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6867rexlimdva 3215 . . . . . . . . . . . 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 402 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
7170impr 444 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7271adantrrl 706 . . . . . . . 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 231 . . . . . . 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 3149 . . . . . 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 4840 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (𝑦 ≼ ω ↔ (𝑡t 𝐴) ≼ ω))
76 rexeq 3324 . . . . . . . . . 10 (𝑦 = (𝑡t 𝐴) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
7776imbi2d 331 . . . . . . . . 9 (𝑦 = (𝑡t 𝐴) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7877ralbidv 3170 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7975, 78anbi12d 618 . . . . . . 7 (𝑦 = (𝑡t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
8079rspcev 3498 . . . . . 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 856 . . . . 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 3219 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
838, 82syldan 581 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8483ralrimiva 3150 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
85 eqid 2802 . . 3 (𝐽t 𝐴) = (𝐽t 𝐴)
8685is1stc2 21453 . 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 574 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2155  wral 3092  wrex 3093  Vcvv 3387  cin 3762  wss 3763  𝒫 cpw 4345   cuni 4623   class class class wbr 4837  cmpt 4916  ran crn 5306  (class class class)co 6868  ωcom 7289  cdom 8184  t crest 16280  Topctop 20905  1st𝜔c1stc 21448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-isom 6104  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-oadd 7794  df-er 7973  df-map 8088  df-en 8187  df-dom 8188  df-fin 8190  df-fi 8550  df-card 9042  df-acn 9045  df-rest 16282  df-topgen 16303  df-top 20906  df-topon 20923  df-bases 20958  df-1stc 21450
This theorem is referenced by:  lly1stc  21507
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