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Theorem 2ndcsep 21072
Description: A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)
Hypothesis
Ref Expression
2ndcsep.1 𝑋 = 𝐽
Assertion
Ref Expression
2ndcsep (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem 2ndcsep
Dummy variables 𝑓 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 21059 . 2 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2 vex 3176 . . . . . . . . 9 𝑏 ∈ V
3 difss 3699 . . . . . . . . 9 (𝑏 ∖ {∅}) ⊆ 𝑏
4 ssdomg 7887 . . . . . . . . 9 (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
52, 3, 4mp2 9 . . . . . . . 8 (𝑏 ∖ {∅}) ≼ 𝑏
6 simpr 476 . . . . . . . 8 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7 domtr 7895 . . . . . . . 8 (((𝑏 ∖ {∅}) ≼ 𝑏𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
85, 6, 7sylancr 694 . . . . . . 7 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9 eldifsn 4260 . . . . . . . . 9 (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦𝑏𝑦 ≠ ∅))
10 n0 3890 . . . . . . . . . 10 (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧𝑦)
11 elunii 4377 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧 𝑏)
12 simpl 472 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧𝑦)
1311, 12jca 553 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦𝑏) → (𝑧 𝑏𝑧𝑦))
1413expcom 450 . . . . . . . . . . . . 13 (𝑦𝑏 → (𝑧𝑦 → (𝑧 𝑏𝑧𝑦)))
1514eximdv 1833 . . . . . . . . . . . 12 (𝑦𝑏 → (∃𝑧 𝑧𝑦 → ∃𝑧(𝑧 𝑏𝑧𝑦)))
1615imp 444 . . . . . . . . . . 11 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧(𝑧 𝑏𝑧𝑦))
17 df-rex 2902 . . . . . . . . . . 11 (∃𝑧 𝑏𝑧𝑦 ↔ ∃𝑧(𝑧 𝑏𝑧𝑦))
1816, 17sylibr 223 . . . . . . . . . 10 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧 𝑏𝑧𝑦)
1910, 18sylan2b 491 . . . . . . . . 9 ((𝑦𝑏𝑦 ≠ ∅) → ∃𝑧 𝑏𝑧𝑦)
209, 19sylbi 206 . . . . . . . 8 (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 𝑏𝑧𝑦)
2120rgen 2906 . . . . . . 7 𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦
22 vuniex 6852 . . . . . . . 8 𝑏 ∈ V
23 eleq1 2676 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝑧𝑦 ↔ (𝑓𝑦) ∈ 𝑦))
2422, 23axcc4dom 9146 . . . . . . 7 (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
258, 21, 24sylancl 693 . . . . . 6 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
26 frn 5966 . . . . . . . . 9 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ran 𝑓 𝑏)
2726ad2antrl 760 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 𝑏)
28 vex 3176 . . . . . . . . . 10 𝑓 ∈ V
2928rnex 6992 . . . . . . . . 9 ran 𝑓 ∈ V
3029elpw 4114 . . . . . . . 8 (ran 𝑓 ∈ 𝒫 𝑏 ↔ ran 𝑓 𝑏)
3127, 30sylibr 223 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 𝑏)
32 omelon 8426 . . . . . . . . . . 11 ω ∈ On
336adantr 480 . . . . . . . . . . 11 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34 ondomen 8743 . . . . . . . . . . 11 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
3532, 33, 34sylancr 694 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36 ssnum 8745 . . . . . . . . . 10 ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
3735, 3, 36sylancl 693 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38 ffn 5958 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑓 Fn (𝑏 ∖ {∅}))
3938ad2antrl 760 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40 dffn4 6034 . . . . . . . . . 10 (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
4139, 40sylib 207 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42 fodomnum 8763 . . . . . . . . 9 ((𝑏 ∖ {∅}) ∈ dom card → (𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓 → ran 𝑓 ≼ (𝑏 ∖ {∅})))
4337, 41, 42sylc 63 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ (𝑏 ∖ {∅}))
448adantr 480 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45 domtr 7895 . . . . . . . 8 ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
4643, 44, 45syl2anc 691 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47 tgcl 20584 . . . . . . . . . 10 (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
4847ad2antrr 758 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49 unitg 20582 . . . . . . . . . . . 12 (𝑏 ∈ V → (topGen‘𝑏) = 𝑏)
502, 49ax-mp 5 . . . . . . . . . . 11 (topGen‘𝑏) = 𝑏
5150eqcomi 2619 . . . . . . . . . 10 𝑏 = (topGen‘𝑏)
5251clsss3 20673 . . . . . . . . 9 (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
5348, 27, 52syl2anc 691 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
54 ne0i 3880 . . . . . . . . . . . . . . . . . 18 (𝑥𝑦𝑦 ≠ ∅)
5554anim2i 591 . . . . . . . . . . . . . . . . 17 ((𝑦𝑏𝑥𝑦) → (𝑦𝑏𝑦 ≠ ∅))
5655, 9sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑦𝑏𝑥𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57 fnfvelrn 6264 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
5838, 57sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
59 inelcm 3984 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑦) ∈ 𝑦 ∧ (𝑓𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
6059expcom 450 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6158, 60syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6261ex 449 . . . . . . . . . . . . . . . . 17 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (𝑦 ∈ (𝑏 ∖ {∅}) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6362a2d 29 . . . . . . . . . . . . . . . 16 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦 ∈ (𝑏 ∖ {∅}) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6456, 63syl7 72 . . . . . . . . . . . . . . 15 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → ((𝑦𝑏𝑥𝑦) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6564exp4a 631 . . . . . . . . . . . . . 14 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦𝑏 → (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
6665ralimdv2 2944 . . . . . . . . . . . . 13 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦 → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6766imp 444 . . . . . . . . . . . 12 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6867ad2antlr 759 . . . . . . . . . . 11 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69 eqidd 2611 . . . . . . . . . . . 12 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → (topGen‘𝑏) = (topGen‘𝑏))
7051a1i 11 . . . . . . . . . . . 12 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 = (topGen‘𝑏))
71 simplll 794 . . . . . . . . . . . 12 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 ∈ TopBases)
7227adantr 480 . . . . . . . . . . . 12 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ran 𝑓 𝑏)
73 simpr 476 . . . . . . . . . . . 12 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 𝑏)
7469, 70, 71, 72, 73elcls3 20697 . . . . . . . . . . 11 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → (𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓) ↔ ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
7568, 74mpbird 246 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
7675ex 449 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑥 𝑏𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓)))
7776ssrdv 3574 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ⊆ ((cls‘(topGen‘𝑏))‘ran 𝑓))
7853, 77eqssd 3585 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)
79 breq1 4586 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
80 fveq2 6103 . . . . . . . . . 10 (𝑥 = ran 𝑓 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘(topGen‘𝑏))‘ran 𝑓))
8180eqeq1d 2612 . . . . . . . . 9 (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏))
8279, 81anbi12d 743 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)))
8382rspcev 3282 . . . . . . 7 ((ran 𝑓 ∈ 𝒫 𝑏 ∧ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8431, 46, 78, 83syl12anc 1316 . . . . . 6 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8525, 84exlimddv 1850 . . . . 5 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
86 unieq 4380 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 (topGen‘𝑏) = 𝐽)
87 2ndcsep.1 . . . . . . . 8 𝑋 = 𝐽
8886, 51, 873eqtr4g 2669 . . . . . . 7 ((topGen‘𝑏) = 𝐽 𝑏 = 𝑋)
8988pweqd 4113 . . . . . 6 ((topGen‘𝑏) = 𝐽 → 𝒫 𝑏 = 𝒫 𝑋)
90 fveq2 6103 . . . . . . . . 9 ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
9190fveq1d 6105 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
9291, 88eqeq12d 2625 . . . . . . 7 ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
9392anbi2d 736 . . . . . 6 ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9489, 93rexeqbidv 3130 . . . . 5 ((topGen‘𝑏) = 𝐽 → (∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9585, 94syl5ibcom 234 . . . 4 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9695impr 647 . . 3 ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
9796rexlimiva 3010 . 2 (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
981, 97sylbi 206 1 (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108  {csn 4125   cuni 4372   class class class wbr 4583  dom cdm 5038  ran crn 5039  Oncon0 5640   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  ωcom 6957  cdom 7839  cardccrd 8644  topGenctg 15921  Topctop 20517  TopBasesctb 20520  clsccl 20632  2nd𝜔c2ndc 21051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-acn 8651  df-topgen 15927  df-top 20521  df-bases 20522  df-cld 20633  df-ntr 20634  df-cls 20635  df-2ndc 21053
This theorem is referenced by:  met2ndc  22138
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