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Theorem 2ndcomap 22151
 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 21934 . . . . . 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 775 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → 𝜑)
6 bastg 21659 . . . . . . . . . 10 (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏))
76ad2antlr 727 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏))
8 simprr 773 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽)
97, 8sseqtrd 3933 . . . . . . . 8 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏𝐽)
109sselda 3893 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → 𝑥𝐽)
11 2ndcomap.7 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
125, 10, 11syl2anc 588 . . . . . 6 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → (𝐹𝑥) ∈ 𝐾)
1312fmpttd 6871 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏𝐾)
1413frnd 6506 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾)
15 elunii 4804 . . . . . . . . . . 11 ((𝑧𝑘𝑘𝐾) → 𝑧 𝐾)
16 2ndcomap.2 . . . . . . . . . . 11 𝑌 = 𝐾
1715, 16eleqtrrdi 2864 . . . . . . . . . 10 ((𝑧𝑘𝑘𝐾) → 𝑧𝑌)
1817ancoms 463 . . . . . . . . 9 ((𝑘𝐾𝑧𝑘) → 𝑧𝑌)
1918adantl 486 . . . . . . . 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 2856 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝑧 ∈ ran 𝐹)
23 eqid 2759 . . . . . . . . . . 11 𝐽 = 𝐽
2423, 16cnf 21939 . . . . . . . . . 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 6499 . . . . . . . 8 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
28 fvelrnb 6715 . . . . . . . 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 235 . . . . . 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 779 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑘𝐾)
33 cnima 21958 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑘𝐾) → (𝐹𝑘) ∈ 𝐽)
3431, 32, 33syl2anc 588 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑘) ∈ 𝐽)
358adantr 485 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (topGen‘𝑏) = 𝐽)
3634, 35eleqtrrd 2856 . . . . . . . . 9 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑘) ∈ (topGen‘𝑏))
37 simprrl 781 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑡 𝐽)
38 simprrr 782 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) = 𝑧)
39 simprlr 780 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑧𝑘)
4038, 39eqeltrd 2853 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) ∈ 𝑘)
4126ffnd 6500 . . . . . . . . . . . 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 6820 . . . . . . . . . . 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 21658 . . . . . . . . 9 (((𝐹𝑘) ∈ (topGen‘𝑏) ∧ 𝑡 ∈ (𝐹𝑘)) → ∃𝑚𝑏 (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))
4736, 45, 46syl2anc 588 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → ∃𝑚𝑏 (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))
48 simprl 771 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚𝑏)
49 eqid 2759 . . . . . . . . . . 11 (𝐹𝑚) = (𝐹𝑚)
50 imaeq2 5898 . . . . . . . . . . . 12 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
5150rspceeqv 3557 . . . . . . . . . . 11 ((𝑚𝑏 ∧ (𝐹𝑚) = (𝐹𝑚)) → ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥))
5248, 49, 51sylancl 590 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → ∃𝑥𝑏 (𝐹𝑚) = (𝐹𝑥))
5342adantr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝐹 Fn 𝐽)
54 fnfun 6435 . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ (𝐹𝑘))
57 funimass2 6419 . . . . . . . . . . . . 13 ((Fun 𝐹𝑚 ⊆ (𝐹𝑘)) → (𝐹𝑚) ⊆ 𝑘)
5855, 56, 57syl2anc 588 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ⊆ 𝑘)
59 vex 3414 . . . . . . . . . . . 12 𝑘 ∈ V
60 ssexg 5194 . . . . . . . . . . . 12 (((𝐹𝑚) ⊆ 𝑘𝑘 ∈ V) → (𝐹𝑚) ∈ V)
6158, 59, 60sylancl 590 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ∈ V)
62 eqid 2759 . . . . . . . . . . . 12 (𝑥𝑏 ↦ (𝐹𝑥)) = (𝑥𝑏 ↦ (𝐹𝑥))
6362elrnmpt 5798 . . . . . . . . . . 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 260 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)))
6638adantr 485 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑡) = 𝑧)
67 simprrl 781 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑡𝑚)
68 cnvimass 5922 . . . . . . . . . . . . 13 (𝐹𝑘) ⊆ dom 𝐹
6956, 68sstrdi 3905 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ dom 𝐹)
70 funfvima2 6986 . . . . . . . . . . . 12 ((Fun 𝐹𝑚 ⊆ dom 𝐹) → (𝑡𝑚 → (𝐹𝑡) ∈ (𝐹𝑚)))
7155, 69, 70syl2anc 588 . . . . . . . . . . 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 2854 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑧 ∈ (𝐹𝑚))
74 eleq2 2841 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑧𝑤𝑧 ∈ (𝐹𝑚)))
75 sseq1 3918 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑤𝑘 ↔ (𝐹𝑚) ⊆ 𝑘))
7674, 75anbi12d 634 . . . . . . . . . 10 (𝑤 = (𝐹𝑚) → ((𝑧𝑤𝑤𝑘) ↔ (𝑧 ∈ (𝐹𝑚) ∧ (𝐹𝑚) ⊆ 𝑘)))
7776rspcev 3542 . . . . . . . . 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 3216 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8079anassrs 472 . . . . . 6 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8130, 80rexlimddv 3216 . . . . 5 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8281ralrimivva 3121 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
83 basgen2 21682 . . . 4 ((𝐾 ∈ Top ∧ ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾 ∧ ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
844, 14, 82, 83syl3anc 1369 . . 3 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
8584, 4eqeltrd 2853 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
86 tgclb 21663 . . . . 5 (ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases ↔ (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
8785, 86sylibr 237 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases)
88 omelon 9135 . . . . . . 7 ω ∈ On
89 simprl 771 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω)
90 ondomen 9490 . . . . . . 7 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
9188, 89, 90sylancr 591 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card)
9213ffnd 6500 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏)
93 dffn4 6583 . . . . . . 7 ((𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏 ↔ (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
9492, 93sylib 221 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
95 fodomnum 9510 . . . . . 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 8581 . . . . 5 ((ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ 𝑏𝑏 ≼ ω) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω)
9896, 89, 97syl2anc 588 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω)
99 2ndci 22141 . . . 4 ((ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases ∧ ran (𝑥𝑏 ↦ (𝐹𝑥)) ≼ ω) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ 2ndω)
10087, 98, 99syl2anc 588 . . 3 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ 2ndω)
10184, 100eqeltrrd 2854 . 2 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ 2ndω)
102 2ndcomap.3 . . 3 (𝜑𝐽 ∈ 2ndω)
103 is2ndc 22139 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
104102, 103sylib 221 . 2 (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
105101, 104r19.29a 3214 1 (𝜑𝐾 ∈ 2ndω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ⊆ wss 3859  ∪ cuni 4799   class class class wbr 5033   ↦ cmpt 5113  ◡ccnv 5524  dom cdm 5525  ran crn 5526   “ cima 5528  Oncon0 6170  Fun wfun 6330   Fn wfn 6331  ⟶wf 6332  –onto→wfo 6334  ‘cfv 6336  (class class class)co 7151  ωcom 7580   ≼ cdom 8526  cardccrd 9390  topGenctg 16762  Topctop 21586  TopBasesctb 21638   Cn ccn 21917  2ndωc2ndc 22131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9130 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-card 9394  df-acn 9397  df-topgen 16768  df-top 21587  df-topon 21604  df-bases 21639  df-cn 21920  df-2ndc 22133 This theorem is referenced by: (None)
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