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Theorem 2ndcomap 23467
Description: A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypotheses
Ref Expression
2ndcomap.2 𝑌 = 𝐾
2ndcomap.3 (𝜑𝐽 ∈ 2ndω)
2ndcomap.5 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2ndcomap.6 (𝜑 → ran 𝐹 = 𝑌)
2ndcomap.7 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
Assertion
Ref Expression
2ndcomap (𝜑𝐾 ∈ 2ndω)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝜑,𝑥   𝑥,𝐾
Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem 2ndcomap
Dummy variables 𝑘 𝑚 𝑡 𝑤 𝑧 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2ndcomap.5 . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 cntop2 23250 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
31, 2syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
43ad2antrr 726 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ Top)
5 simplll 774 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → 𝜑)
6 bastg 22974 . . . . . . . . . 10 (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏))
76ad2antlr 727 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏))
8 simprr 772 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽)
97, 8sseqtrd 4019 . . . . . . . 8 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏𝐽)
109sselda 3982 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → 𝑥𝐽)
11 2ndcomap.7 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
125, 10, 11syl2anc 584 . . . . . 6 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → (𝐹𝑥) ∈ 𝐾)
1312fmpttd 7134 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏𝐾)
1413frnd 6743 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾)
15 elunii 4911 . . . . . . . . . . 11 ((𝑧𝑘𝑘𝐾) → 𝑧 𝐾)
16 2ndcomap.2 . . . . . . . . . . 11 𝑌 = 𝐾
1715, 16eleqtrrdi 2851 . . . . . . . . . 10 ((𝑧𝑘𝑘𝐾) → 𝑧𝑌)
1817ancoms 458 . . . . . . . . 9 ((𝑘𝐾𝑧𝑘) → 𝑧𝑌)
1918adantl 481 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝑧𝑌)
20 2ndcomap.6 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝑌)
2120ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ran 𝐹 = 𝑌)
2219, 21eleqtrrd 2843 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝑧 ∈ ran 𝐹)
23 eqid 2736 . . . . . . . . . . 11 𝐽 = 𝐽
2423, 16cnf 23255 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
251, 24syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽𝑌)
2625ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝐹: 𝐽𝑌)
27 ffn 6735 . . . . . . . 8 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
28 fvelrnb 6968 . . . . . . . 8 (𝐹 Fn 𝐽 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 𝐽(𝐹𝑡) = 𝑧))
2926, 27, 283syl 18 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 𝐽(𝐹𝑡) = 𝑧))
3022, 29mpbid 232 . . . . . 6 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ∃𝑡 𝐽(𝐹𝑡) = 𝑧)
311ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝐹 ∈ (𝐽 Cn 𝐾))
32 simprll 778 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑘𝐾)
33 cnima 23274 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑘𝐾) → (𝐹𝑘) ∈ 𝐽)
3431, 32, 33syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑘) ∈ 𝐽)
358adantr 480 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (topGen‘𝑏) = 𝐽)
3634, 35eleqtrrd 2843 . . . . . . . . 9 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑘) ∈ (topGen‘𝑏))
37 simprrl 780 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑡 𝐽)
38 simprrr 781 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) = 𝑧)
39 simprlr 779 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑧𝑘)
4038, 39eqeltrd 2840 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) ∈ 𝑘)
4126ffnd 6736 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝐹 Fn 𝐽)
4241adantrr 717 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝐹 Fn 𝐽)
43 elpreima 7077 . . . . . . . . . . 11 (𝐹 Fn 𝐽 → (𝑡 ∈ (𝐹𝑘) ↔ (𝑡 𝐽 ∧ (𝐹𝑡) ∈ 𝑘)))
4442, 43syl 17 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝑡 ∈ (𝐹𝑘) ↔ (𝑡 𝐽 ∧ (𝐹𝑡) ∈ 𝑘)))
4537, 40, 44mpbir2and 713 . . . . . . . . 9 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑡 ∈ (𝐹𝑘))
46 tg2 22973 . . . . . . . . 9 (((𝐹𝑘) ∈ (topGen‘𝑏) ∧ 𝑡 ∈ (𝐹𝑘)) → ∃𝑚𝑏 (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))
4736, 45, 46syl2anc 584 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → ∃𝑚𝑏 (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))
48 simprl 770 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚𝑏)
49 eqid 2736 . . . . . . . . . . 11 (𝐹𝑚) = (𝐹𝑚)
50 imaeq2 6073 . . . . . . . . . . . 12 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
5150rspceeqv 3644 . . . . . . . . . . 11 ((𝑚𝑏 ∧ (𝐹𝑚) = (𝐹𝑚)) → ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥))
5248, 49, 51sylancl 586 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥))
5342adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝐹 Fn 𝐽)
54 fnfun 6667 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐽 → Fun 𝐹)
5553, 54syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → Fun 𝐹)
56 simprrr 781 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ (𝐹𝑘))
57 funimass2 6648 . . . . . . . . . . . . 13 ((Fun 𝐹𝑚 ⊆ (𝐹𝑘)) → (𝐹𝑚) ⊆ 𝑘)
5855, 56, 57syl2anc 584 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ⊆ 𝑘)
59 vex 3483 . . . . . . . . . . . 12 𝑘 ∈ V
60 ssexg 5322 . . . . . . . . . . . 12 (((𝐹𝑚) ⊆ 𝑘𝑘 ∈ V) → (𝐹𝑚) ∈ V)
6158, 59, 60sylancl 586 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ∈ V)
62 eqid 2736 . . . . . . . . . . . 12 (𝑥𝑏 ↦ (𝐹𝑥)) = (𝑥𝑏 ↦ (𝐹𝑥))
6362elrnmpt 5968 . . . . . . . . . . 11 ((𝐹𝑚) ∈ V → ((𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)) ↔ ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥)))
6461, 63syl 17 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → ((𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)) ↔ ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥)))
6552, 64mpbird 257 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)))
6638adantr 480 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑡) = 𝑧)
67 simprrl 780 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑡𝑚)
68 cnvimass 6099 . . . . . . . . . . . . 13 (𝐹𝑘) ⊆ dom 𝐹
6956, 68sstrdi 3995 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ dom 𝐹)
70 funfvima2 7252 . . . . . . . . . . . 12 ((Fun 𝐹𝑚 ⊆ dom 𝐹) → (𝑡𝑚 → (𝐹𝑡) ∈ (𝐹𝑚)))
7155, 69, 70syl2anc 584 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝑡𝑚 → (𝐹𝑡) ∈ (𝐹𝑚)))
7267, 71mpd 15 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑡) ∈ (𝐹𝑚))
7366, 72eqeltrrd 2841 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑧 ∈ (𝐹𝑚))
74 eleq2 2829 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑧𝑤𝑧 ∈ (𝐹𝑚)))
75 sseq1 4008 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑤𝑘 ↔ (𝐹𝑚) ⊆ 𝑘))
7674, 75anbi12d 632 . . . . . . . . . 10 (𝑤 = (𝐹𝑚) → ((𝑧𝑤𝑤𝑘) ↔ (𝑧 ∈ (𝐹𝑚) ∧ (𝐹𝑚) ⊆ 𝑘)))
7776rspcev 3621 . . . . . . . . 9 (((𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)) ∧ (𝑧 ∈ (𝐹𝑚) ∧ (𝐹𝑚) ⊆ 𝑘)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
7865, 73, 58, 77syl12anc 836 . . . . . . . 8 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
7947, 78rexlimddv 3160 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8079anassrs 467 . . . . . 6 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8130, 80rexlimddv 3160 . . . . 5 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8281ralrimivva 3201 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
83 basgen2 22997 . . . 4 ((𝐾 ∈ Top ∧ ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾 ∧ ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
844, 14, 82, 83syl3anc 1372 . . 3 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
8584, 4eqeltrd 2840 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
86 tgclb 22978 . . . . 5 (ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases ↔ (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
8785, 86sylibr 234 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases)
88 omelon 9687 . . . . . . 7 ω ∈ On
89 simprl 770 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω)
90 ondomen 10078 . . . . . . 7 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
9188, 89, 90sylancr 587 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card)
9213ffnd 6736 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏)
93 dffn4 6825 . . . . . . 7 ((𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏 ↔ (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
9492, 93sylib 218 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
95 fodomnum 10098 . . . . . 6 (𝑏 ∈ dom card → ((𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ 𝑏))
9691, 94, 95sylc 65 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ 𝑏)
97 domtr 9048 . . . . 5 ((ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ 𝑏𝑏 ≼ ω) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω)
9896, 89, 97syl2anc 584 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω)
99 2ndci 23457 . . . 4 ((ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases ∧ ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ 2ndω)
10087, 98, 99syl2anc 584 . . 3 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ 2ndω)
10184, 100eqeltrrd 2841 . 2 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ 2ndω)
102 2ndcomap.3 . . 3 (𝜑𝐽 ∈ 2ndω)
103 is2ndc 23455 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
104102, 103sylib 218 . 2 (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
105101, 104r19.29a 3161 1 (𝜑𝐾 ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  wss 3950   cuni 4906   class class class wbr 5142  cmpt 5224  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  ωcom 7888  cdom 8984  cardccrd 9976  topGenctg 17483  Topctop 22900  TopBasesctb 22953   Cn ccn 23233  2ndωc2ndc 23447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-card 9980  df-acn 9983  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cn 23236  df-2ndc 23449
This theorem is referenced by: (None)
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