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Theorem 2ndcomap 21994
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 21777 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
31, 2syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
43ad2antrr 722 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ Top)
5 simplll 771 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥𝑏) → 𝜑)
6 bastg 21502 . . . . . . . . . 10 (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏))
76ad2antlr 723 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏))
8 simprr 769 . . . . . . . . 9 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽)
97, 8sseqtrd 4004 . . . . . . . 8 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏𝐽)
109sselda 3964 . . . . . . 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 6871 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏𝐾)
1413frnd 6514 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾)
15 elunii 4835 . . . . . . . . . . 11 ((𝑧𝑘𝑘𝐾) → 𝑧 𝐾)
16 2ndcomap.2 . . . . . . . . . . 11 𝑌 = 𝐾
1715, 16eleqtrrdi 2921 . . . . . . . . . 10 ((𝑧𝑘𝑘𝐾) → 𝑧𝑌)
1817ancoms 459 . . . . . . . . 9 ((𝑘𝐾𝑧𝑘) → 𝑧𝑌)
1918adantl 482 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝑧𝑌)
20 2ndcomap.6 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝑌)
2120ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ran 𝐹 = 𝑌)
2219, 21eleqtrrd 2913 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝑧 ∈ ran 𝐹)
23 eqid 2818 . . . . . . . . . . 11 𝐽 = 𝐽
2423, 16cnf 21782 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
251, 24syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽𝑌)
2625ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝐹: 𝐽𝑌)
27 ffn 6507 . . . . . . . 8 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
28 fvelrnb 6719 . . . . . . . 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 233 . . . . . 6 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ∃𝑡 𝐽(𝐹𝑡) = 𝑧)
311ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝐹 ∈ (𝐽 Cn 𝐾))
32 simprll 775 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑘𝐾)
33 cnima 21801 . . . . . . . . . . 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 481 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (topGen‘𝑏) = 𝐽)
3634, 35eleqtrrd 2913 . . . . . . . . 9 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑘) ∈ (topGen‘𝑏))
37 simprrl 777 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑡 𝐽)
38 simprrr 778 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) = 𝑧)
39 simprlr 776 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑧𝑘)
4038, 39eqeltrd 2910 . . . . . . . . . 10 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → (𝐹𝑡) ∈ 𝑘)
4126ffnd 6508 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → 𝐹 Fn 𝐽)
4241adantrr 713 . . . . . . . . . . 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 709 . . . . . . . . 9 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → 𝑡 ∈ (𝐹𝑘))
46 tg2 21501 . . . . . . . . 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 767 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚𝑏)
49 eqid 2818 . . . . . . . . . . 11 (𝐹𝑚) = (𝐹𝑚)
50 imaeq2 5918 . . . . . . . . . . . 12 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
5150rspceeqv 3635 . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝐹 Fn 𝐽)
54 fnfun 6446 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ (𝐹𝑘))
57 funimass2 6430 . . . . . . . . . . . . 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 3495 . . . . . . . . . . . 12 𝑘 ∈ V
60 ssexg 5218 . . . . . . . . . . . 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 2818 . . . . . . . . . . . 12 (𝑥𝑏 ↦ (𝐹𝑥)) = (𝑥𝑏 ↦ (𝐹𝑥))
6362elrnmpt 5821 . . . . . . . . . . 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 258 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)))
6638adantr 481 . . . . . . . . . 10 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → (𝐹𝑡) = 𝑧)
67 simprrl 777 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑡𝑚)
68 cnvimass 5942 . . . . . . . . . . . . 13 (𝐹𝑘) ⊆ dom 𝐹
6956, 68sstrdi 3976 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑚 ⊆ dom 𝐹)
70 funfvima2 6984 . . . . . . . . . . . 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 2911 . . . . . . . . 9 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → 𝑧 ∈ (𝐹𝑚))
74 eleq2 2898 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑧𝑤𝑧 ∈ (𝐹𝑚)))
75 sseq1 3989 . . . . . . . . . . 11 (𝑤 = (𝐹𝑚) → (𝑤𝑘 ↔ (𝐹𝑚) ⊆ 𝑘))
7674, 75anbi12d 630 . . . . . . . . . 10 (𝑤 = (𝐹𝑚) → ((𝑧𝑤𝑤𝑘) ↔ (𝑧 ∈ (𝐹𝑚) ∧ (𝐹𝑚) ⊆ 𝑘)))
7776rspcev 3620 . . . . . . . . 9 (((𝐹𝑚) ∈ ran (𝑥𝑏 ↦ (𝐹𝑥)) ∧ (𝑧 ∈ (𝐹𝑚) ∧ (𝐹𝑚) ⊆ 𝑘)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
7865, 73, 58, 77syl12anc 832 . . . . . . . 8 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) ∧ (𝑚𝑏 ∧ (𝑡𝑚𝑚 ⊆ (𝐹𝑘)))) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
7947, 78rexlimddv 3288 . . . . . . 7 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘𝐾𝑧𝑘) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧))) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8079anassrs 468 . . . . . 6 (((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) ∧ (𝑡 𝐽 ∧ (𝐹𝑡) = 𝑧)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8130, 80rexlimddv 3288 . . . . 5 ((((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘𝐾𝑧𝑘)) → ∃𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
8281ralrimivva 3188 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘))
83 basgen2 21525 . . . 4 ((𝐾 ∈ Top ∧ ran (𝑥𝑏 ↦ (𝐹𝑥)) ⊆ 𝐾 ∧ ∀𝑘𝐾𝑧𝑘𝑤 ∈ ran (𝑥𝑏 ↦ (𝐹𝑥))(𝑧𝑤𝑤𝑘)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
844, 14, 82, 83syl3anc 1363 . . 3 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) = 𝐾)
8584, 4eqeltrd 2910 . . . . 5 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
86 tgclb 21506 . . . . 5 (ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases ↔ (topGen‘ran (𝑥𝑏 ↦ (𝐹𝑥))) ∈ Top)
8785, 86sylibr 235 . . . 4 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥𝑏 ↦ (𝐹𝑥)) ∈ TopBases)
88 omelon 9097 . . . . . . 7 ω ∈ On
89 simprl 767 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω)
90 ondomen 9451 . . . . . . 7 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
9188, 89, 90sylancr 587 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card)
9213ffnd 6508 . . . . . . 7 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏)
93 dffn4 6589 . . . . . . 7 ((𝑥𝑏 ↦ (𝐹𝑥)) Fn 𝑏 ↔ (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
9492, 93sylib 219 . . . . . 6 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥𝑏 ↦ (𝐹𝑥)):𝑏onto→ran (𝑥𝑏 ↦ (𝐹𝑥)))
95 fodomnum 9471 . . . . . 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 8550 . . . . 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 21984 . . . 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 2911 . 2 (((𝜑𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ 2ndω)
102 2ndcomap.3 . . 3 (𝜑𝐽 ∈ 2ndω)
103 is2ndc 21982 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
104102, 103sylib 219 . 2 (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
105101, 104r19.29a 3286 1 (𝜑𝐾 ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933   cuni 4830   class class class wbr 5057  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  Oncon0 6184  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7145  ωcom 7569  cdom 8495  cardccrd 9352  topGenctg 16699  Topctop 21429  TopBasesctb 21481   Cn ccn 21760  2ndωc2ndc 21974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-card 9356  df-acn 9359  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-2ndc 21976
This theorem is referenced by: (None)
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