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Theorem 1stcrest 22804
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 22794 . . 3 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2 resttop 22511 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 580 . 2 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 eqid 2736 . . . . . . . 8 𝐽 = 𝐽
54restuni2 22518 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
61, 5sylan 580 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
76eleq2d 2823 . . . . 5 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → (𝑥 ∈ (𝐴 𝐽) ↔ 𝑥 (𝐽t 𝐴)))
87biimpar 478 . . . 4 (((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → 𝑥 ∈ (𝐴 𝐽))
9 simpl 483 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → 𝐽 ∈ 1stω)
10 elinel2 4156 . . . . . 6 (𝑥 ∈ (𝐴 𝐽) → 𝑥 𝐽)
1141stcclb 22795 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑥 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
129, 10, 11syl2an 596 . . . . 5 (((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
13 simplll 773 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ 1stω)
14 elpwi 4567 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐽𝑡𝐽)
1514ad2antrl 726 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡𝐽)
16 ssrest 22527 . . . . . . . 8 ((𝐽 ∈ 1stω ∧ 𝑡𝐽) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
1713, 15, 16syl2anc 584 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
18 ovex 7390 . . . . . . . 8 (𝐽t 𝐴) ∈ V
1918elpw2 5302 . . . . . . 7 ((𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴) ↔ (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
2017, 19sylibr 233 . . . . . 6 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴))
21 vex 3449 . . . . . . . 8 𝑡 ∈ V
22 simpllr 774 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐴𝑉)
23 restval 17308 . . . . . . . 8 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
2421, 22, 23sylancr 587 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
25 simprrl 779 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡 ≼ ω)
26 1stcrestlem 22803 . . . . . . . 8 (𝑡 ≼ ω → ran (𝑣𝑡 ↦ (𝑣𝐴)) ≼ ω)
2725, 26syl 17 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → ran (𝑣𝑡 ↦ (𝑣𝐴)) ≼ ω)
2824, 27eqbrtrd 5127 . . . . . 6 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ≼ ω)
291ad3antrrr 728 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ Top)
30 elrest 17309 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
3129, 22, 30syl2anc 584 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
32 r19.29 3117 . . . . . . . . . . . 12 ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → ∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)))
33 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝑥𝐴)
3433a1d 25 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥𝐴))
3534ancld 551 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦 → (𝑥𝑦𝑥𝐴)))
36 elin 3926 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
3735, 36syl6ibr 251 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥 ∈ (𝑦𝐴)))
38 ssrin 4193 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴))
3937, 38anim12d1 610 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → ((𝑥𝑦𝑦𝑎) → (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
4039reximdv 3167 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
41 vex 3449 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
4241inex1 5274 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐴) ∈ V
4342a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑦𝑡) → (𝑦𝐴) ∈ V)
44 simp-4r 782 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝐴𝑉)
45 elrest 17309 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
4621, 44, 45sylancr 587 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
47 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑥𝑤𝑥 ∈ (𝑦𝐴)))
48 sseq1 3969 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑤 ⊆ (𝑎𝐴) ↔ (𝑦𝐴) ⊆ (𝑎𝐴)))
4947, 48anbi12d 631 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (𝑦𝐴) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5049adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑤 = (𝑦𝐴)) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5143, 46, 50rexxfr2d 5366 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5240, 51sylibrd 258 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
5352expr 457 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (𝑥𝐴 → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
5453com23 86 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → (𝑥𝐴 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
5554imim2d 57 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (𝑥𝑎 → (𝑥𝐴 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))))
5655imp4b 422 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
57 eleq2 2826 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → (𝑥𝑧𝑥 ∈ (𝑎𝐴)))
58 elin 3926 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑎𝐴) ↔ (𝑥𝑎𝑥𝐴))
5957, 58bitrdi 286 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (𝑥𝑧 ↔ (𝑥𝑎𝑥𝐴)))
60 sseq2 3970 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑎𝐴) → (𝑤𝑧𝑤 ⊆ (𝑎𝐴)))
6160anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6261rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6359, 62imbi12d 344 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐴) → ((𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)) ↔ ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
6456, 63syl5ibrcom 246 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → (𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6564expimpd 454 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6665rexlimdva 3152 . . . . . . . . . . . 12 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6732, 66syl5 34 . . . . . . . . . . 11 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6867expd 416 . . . . . . . . . 10 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
6968impr 455 . . . . . . . . 9 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7069adantrrl 722 . . . . . . . 8 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7131, 70sylbid 239 . . . . . . 7 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑧 ∈ (𝐽t 𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7271ralrimiv 3142 . . . . . 6 ((((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
73 breq1 5108 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (𝑦 ≼ ω ↔ (𝑡t 𝐴) ≼ ω))
74 rexeq 3310 . . . . . . . . . 10 (𝑦 = (𝑡t 𝐴) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
7574imbi2d 340 . . . . . . . . 9 (𝑦 = (𝑡t 𝐴) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7675ralbidv 3174 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7773, 76anbi12d 631 . . . . . . 7 (𝑦 = (𝑡t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
7877rspcev 3581 . . . . . 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))))
7920, 28, 72, 78syl12anc 835 . . . . 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))))
8012, 79rexlimddv 3158 . . . 4 (((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
818, 80syldan 591 . . 3 (((𝐽 ∈ 1stω ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8281ralrimiva 3143 . 2 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
83 eqid 2736 . . 3 (𝐽t 𝐴) = (𝐽t 𝐴)
8483is1stc2 22793 . 2 ((𝐽t 𝐴) ∈ 1stω ↔ ((𝐽t 𝐴) ∈ Top ∧ ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
853, 82, 84sylanbrc 583 1 ((𝐽 ∈ 1stω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1stω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  cmpt 5188  ran crn 5634  (class class class)co 7357  ωcom 7802  cdom 8881  t crest 17302  Topctop 22242  1stωc1stc 22788
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-card 9875  df-acn 9878  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-1stc 22790
This theorem is referenced by:  lly1stc  22847
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