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Theorem 1stcrest 21179
 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 21169 . . 3 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 resttop 20887 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 488 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 eqid 2621 . . . . . . . 8 𝐽 = 𝐽
54restuni2 20894 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
61, 5sylan 488 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
76eleq2d 2684 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝑥 ∈ (𝐴 𝐽) ↔ 𝑥 (𝐽t 𝐴)))
87biimpar 502 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → 𝑥 ∈ (𝐴 𝐽))
9 simpl 473 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → 𝐽 ∈ 1st𝜔)
10 inss2 3817 . . . . . . 7 (𝐴 𝐽) ⊆ 𝐽
1110sseli 3583 . . . . . 6 (𝑥 ∈ (𝐴 𝐽) → 𝑥 𝐽)
1241stcclb 21170 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
139, 11, 12syl2an 494 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
14 simplll 797 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ 1st𝜔)
15 elpwi 4145 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐽𝑡𝐽)
1615ad2antrl 763 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡𝐽)
17 ssrest 20903 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑡𝐽) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
1814, 16, 17syl2anc 692 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
19 ovex 6638 . . . . . . . 8 (𝐽t 𝐴) ∈ V
2019elpw2 4793 . . . . . . 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 3192 . . . . . . . 8 𝑡 ∈ V
23 simpllr 798 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐴𝑉)
24 restval 16019 . . . . . . . 8 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
2522, 23, 24sylancr 694 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
26 simprrl 803 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡 ≼ ω)
27 1stcrestlem 21178 . . . . . . . 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 4640 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ≼ ω)
301ad3antrrr 765 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ Top)
31 elrest 16020 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
3230, 23, 31syl2anc 692 . . . . . . . 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 3066 . . . . . . . . . . . 12 ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → ∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)))
34 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦 → (𝑥𝑦𝑥𝐴)))
37 elin 3779 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
3836, 37syl6ibr 242 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥 ∈ (𝑦𝐴)))
39 ssrin 3821 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴))
4039a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴)))
4138, 40anim12d 585 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → ((𝑥𝑦𝑦𝑎) → (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
4241reximdv 3011 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
43 vex 3192 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
4443inex1 4764 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐴) ∈ V
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑦𝑡) → (𝑦𝐴) ∈ V)
46 simp-4r 806 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝐴𝑉)
47 elrest 16020 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
4822, 46, 47sylancr 694 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
49 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑥𝑤𝑥 ∈ (𝑦𝐴)))
50 sseq1 3610 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑤 ⊆ (𝑎𝐴) ↔ (𝑦𝐴) ⊆ (𝑎𝐴)))
5149, 50anbi12d 746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (𝑦𝐴) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5251adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑤 = (𝑦𝐴)) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5345, 48, 52rexxfr2d 4848 . . . . . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
59 eleq2 2687 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → (𝑥𝑧𝑥 ∈ (𝑎𝐴)))
60 elin 3779 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑎𝐴) ↔ (𝑥𝑎𝑥𝐴))
6159, 60syl6bb 276 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (𝑥𝑧 ↔ (𝑥𝑎𝑥𝐴)))
62 sseq2 3611 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑎𝐴) → (𝑤𝑧𝑤 ⊆ (𝑎𝐴)))
6362anbi2d 739 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6463rexbidv 3046 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6561, 64imbi12d 334 . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6867rexlimdva 3025 . . . . . . . . . . . 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 452 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
7170impr 648 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7271adantrrl 759 . . . . . . . 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 2960 . . . . . 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 4621 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (𝑦 ≼ ω ↔ (𝑡t 𝐴) ≼ ω))
76 rexeq 3131 . . . . . . . . . 10 (𝑦 = (𝑡t 𝐴) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
7776imbi2d 330 . . . . . . . . 9 (𝑦 = (𝑡t 𝐴) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7877ralbidv 2981 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7975, 78anbi12d 746 . . . . . . 7 (𝑦 = (𝑡t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
8079rspcev 3298 . . . . . 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 1321 . . . . 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 3029 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
838, 82syldan 487 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8483ralrimiva 2961 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
85 eqid 2621 . . 3 (𝐽t 𝐴) = (𝐽t 𝐴)
8685is1stc2 21168 . 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 697 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ∩ cin 3558   ⊆ wss 3559  𝒫 cpw 4135  ∪ cuni 4407   class class class wbr 4618   ↦ cmpt 4678  ran crn 5080  (class class class)co 6610  ωcom 7019   ≼ cdom 7905   ↾t crest 16013  Topctop 20630  1st𝜔c1stc 21163 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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 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-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-fin 7911  df-fi 8269  df-card 8717  df-acn 8720  df-rest 16015  df-topgen 16036  df-top 20631  df-topon 20648  df-bases 20674  df-1stc 21165 This theorem is referenced by:  lly1stc  21222
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