MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stcrest Structured version   Visualization version   GIF version

Theorem 1stcrest 21665
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 21655 . . 3 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 resttop 21372 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 575 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 eqid 2778 . . . . . . . 8 𝐽 = 𝐽
54restuni2 21379 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
61, 5sylan 575 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
76eleq2d 2845 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝑥 ∈ (𝐴 𝐽) ↔ 𝑥 (𝐽t 𝐴)))
87biimpar 471 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → 𝑥 ∈ (𝐴 𝐽))
9 simpl 476 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → 𝐽 ∈ 1st𝜔)
10 inss2 4054 . . . . . . 7 (𝐴 𝐽) ⊆ 𝐽
1110sseli 3817 . . . . . 6 (𝑥 ∈ (𝐴 𝐽) → 𝑥 𝐽)
1241stcclb 21656 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
139, 11, 12syl2an 589 . . . . 5 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))
14 simplll 765 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ 1st𝜔)
15 elpwi 4389 . . . . . . . . 9 (𝑡 ∈ 𝒫 𝐽𝑡𝐽)
1615ad2antrl 718 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡𝐽)
17 ssrest 21388 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑡𝐽) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
1814, 16, 17syl2anc 579 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
19 ovex 6954 . . . . . . . 8 (𝐽t 𝐴) ∈ V
2019elpw2 5062 . . . . . . 7 ((𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴) ↔ (𝑡t 𝐴) ⊆ (𝐽t 𝐴))
2118, 20sylibr 226 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ∈ 𝒫 (𝐽t 𝐴))
22 vex 3401 . . . . . . . 8 𝑡 ∈ V
23 simpllr 766 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐴𝑉)
24 restval 16473 . . . . . . . 8 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
2522, 23, 24sylancr 581 . . . . . . 7 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) = ran (𝑣𝑡 ↦ (𝑣𝐴)))
26 simprrl 771 . . . . . . . 8 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝑡 ≼ ω)
27 1stcrestlem 21664 . . . . . . . 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 4908 . . . . . 6 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → (𝑡t 𝐴) ≼ ω)
301ad3antrrr 720 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))))) → 𝐽 ∈ Top)
31 elrest 16474 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑎𝐽 𝑧 = (𝑎𝐴)))
3230, 23, 31syl2anc 579 . . . . . . . 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 3258 . . . . . . . . . . . 12 ((∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ ∃𝑎𝐽 𝑧 = (𝑎𝐴)) → ∃𝑎𝐽 ((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)))
34 simprr 763 . . . . . . . . . . . . . . . . . . . . . . . . 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 546 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦 → (𝑥𝑦𝑥𝐴)))
37 elin 4019 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (𝑦𝐴) ↔ (𝑥𝑦𝑥𝐴))
3836, 37syl6ibr 244 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑥𝑦𝑥 ∈ (𝑦𝐴)))
39 ssrin 4058 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴))
4039a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑦𝑎 → (𝑦𝐴) ⊆ (𝑎𝐴)))
4138, 40anim12d 602 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → ((𝑥𝑦𝑦𝑎) → (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
4241reximdv 3197 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
43 vex 3401 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ V
4443inex1 5036 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐴) ∈ V
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑦𝑡) → (𝑦𝐴) ∈ V)
46 simp-4r 774 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → 𝐴𝑉)
47 elrest 16474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ V ∧ 𝐴𝑉) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
4822, 46, 47sylancr 581 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (𝑤 ∈ (𝑡t 𝐴) ↔ ∃𝑦𝑡 𝑤 = (𝑦𝐴)))
49 eleq2 2848 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑥𝑤𝑥 ∈ (𝑦𝐴)))
50 sseq1 3845 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑦𝐴) → (𝑤 ⊆ (𝑎𝐴) ↔ (𝑦𝐴) ⊆ (𝑎𝐴)))
5149, 50anbi12d 624 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = (𝑦𝐴) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5251adantl 475 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) ∧ 𝑤 = (𝑦𝐴)) → ((𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5345, 48, 52rexxfr2d 5123 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)) ↔ ∃𝑦𝑡 (𝑥 ∈ (𝑦𝐴) ∧ (𝑦𝐴) ⊆ (𝑎𝐴))))
5442, 53sylibrd 251 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎𝐽𝑥𝐴)) → (∃𝑦𝑡 (𝑥𝑦𝑦𝑎) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
5554expr 450 . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
59 eleq2 2848 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → (𝑥𝑧𝑥 ∈ (𝑎𝐴)))
60 elin 4019 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑎𝐴) ↔ (𝑥𝑎𝑥𝐴))
6159, 60syl6bb 279 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (𝑥𝑧 ↔ (𝑥𝑎𝑥𝐴)))
62 sseq2 3846 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑎𝐴) → (𝑤𝑧𝑤 ⊆ (𝑎𝐴)))
6362anbi2d 622 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎𝐴) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6463rexbidv 3237 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎𝐴) → (∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴))))
6561, 64imbi12d 336 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐴) → ((𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)) ↔ ((𝑥𝑎𝑥𝐴) → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤 ⊆ (𝑎𝐴)))))
6658, 65syl5ibrcom 239 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) ∧ (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎))) → (𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6766expimpd 447 . . . . . . . . . . . . 13 (((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎𝐽) → (((𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) ∧ 𝑧 = (𝑎𝐴)) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
6867rexlimdva 3213 . . . . . . . . . . . 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 406 . . . . . . . . . 10 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
7170impr 448 . . . . . . . . 9 ((((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎𝐽 (𝑥𝑎 → ∃𝑦𝑡 (𝑥𝑦𝑦𝑎)))) → (∃𝑎𝐽 𝑧 = (𝑎𝐴) → (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7271adantrrl 714 . . . . . . . 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 232 . . . . . . 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 3147 . . . . . 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 4889 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (𝑦 ≼ ω ↔ (𝑡t 𝐴) ≼ ω))
76 rexeq 3331 . . . . . . . . . 10 (𝑦 = (𝑡t 𝐴) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))
7776imbi2d 332 . . . . . . . . 9 (𝑦 = (𝑡t 𝐴) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7877ralbidv 3168 . . . . . . . 8 (𝑦 = (𝑡t 𝐴) → (∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧))))
7975, 78anbi12d 624 . . . . . . 7 (𝑦 = (𝑡t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ ((𝑡t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤 ∈ (𝑡t 𝐴)(𝑥𝑤𝑤𝑧)))))
8079rspcev 3511 . . . . . 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 827 . . . . 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 3218 . . . 4 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 ∈ (𝐴 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
838, 82syldan 585 . . 3 (((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) ∧ 𝑥 (𝐽t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8483ralrimiva 3148 . 2 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → ∀𝑥 (𝐽t 𝐴)∃𝑦 ∈ 𝒫 (𝐽t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽t 𝐴)(𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
85 eqid 2778 . . 3 (𝐽t 𝐴) = (𝐽t 𝐴)
8685is1stc2 21654 . 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 578 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  Vcvv 3398  cin 3791  wss 3792  𝒫 cpw 4379   cuni 4671   class class class wbr 4886  cmpt 4965  ran crn 5356  (class class class)co 6922  ωcom 7343  cdom 8239  t crest 16467  Topctop 21105  1st𝜔c1stc 21649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-fin 8245  df-fi 8605  df-card 9098  df-acn 9101  df-rest 16469  df-topgen 16490  df-top 21106  df-topon 21123  df-bases 21158  df-1stc 21651
This theorem is referenced by:  lly1stc  21708
  Copyright terms: Public domain W3C validator