| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 2ndcomap.5 | . . . . . 6
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | 
| 2 |  | cntop2 23250 | . . . . . 6
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | 
| 3 | 1, 2 | syl 17 | . . . . 5
⊢ (𝜑 → 𝐾 ∈ Top) | 
| 4 | 3 | ad2antrr 726 | . . . 4
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈ Top) | 
| 5 |  | simplll 774 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → 𝜑) | 
| 6 |  | bastg 22974 | . . . . . . . . . 10
⊢ (𝑏 ∈ TopBases → 𝑏 ⊆ (topGen‘𝑏)) | 
| 7 | 6 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ (topGen‘𝑏)) | 
| 8 |  | simprr 772 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘𝑏) = 𝐽) | 
| 9 | 7, 8 | sseqtrd 4019 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ⊆ 𝐽) | 
| 10 | 9 | sselda 3982 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ 𝐽) | 
| 11 |  | 2ndcomap.7 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) | 
| 12 | 5, 10, 11 | syl2anc 584 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ 𝑥 ∈ 𝑏) → (𝐹 “ 𝑥) ∈ 𝐾) | 
| 13 | 12 | fmpttd 7134 | . . . . 5
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏⟶𝐾) | 
| 14 | 13 | frnd 6743 | . . . 4
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ⊆ 𝐾) | 
| 15 |  | elunii 4911 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ ∪ 𝐾) | 
| 16 |  | 2ndcomap.2 | . . . . . . . . . . 11
⊢ 𝑌 = ∪
𝐾 | 
| 17 | 15, 16 | eleqtrrdi 2851 | . . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑘 ∧ 𝑘 ∈ 𝐾) → 𝑧 ∈ 𝑌) | 
| 18 | 17 | ancoms 458 | . . . . . . . . 9
⊢ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) → 𝑧 ∈ 𝑌) | 
| 19 | 18 | adantl 481 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝑧 ∈ 𝑌) | 
| 20 |  | 2ndcomap.6 | . . . . . . . . 9
⊢ (𝜑 → ran 𝐹 = 𝑌) | 
| 21 | 20 | ad3antrrr 730 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ran 𝐹 = 𝑌) | 
| 22 | 19, 21 | eleqtrrd 2843 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝑧 ∈ ran 𝐹) | 
| 23 |  | eqid 2736 | . . . . . . . . . . 11
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 24 | 23, 16 | cnf 23255 | . . . . . . . . . 10
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌) | 
| 25 | 1, 24 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌) | 
| 26 | 25 | ad3antrrr 730 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹:∪ 𝐽⟶𝑌) | 
| 27 |  | ffn 6735 | . . . . . . . 8
⊢ (𝐹:∪
𝐽⟶𝑌 → 𝐹 Fn ∪ 𝐽) | 
| 28 |  | fvelrnb 6968 | . . . . . . . 8
⊢ (𝐹 Fn ∪
𝐽 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧)) | 
| 29 | 26, 27, 28 | 3syl 18 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧)) | 
| 30 | 22, 29 | mpbid 232 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ∃𝑡 ∈ ∪ 𝐽(𝐹‘𝑡) = 𝑧) | 
| 31 | 1 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 ∈ (𝐽 Cn 𝐾)) | 
| 32 |  | simprll 778 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑘 ∈ 𝐾) | 
| 33 |  | cnima 23274 | . . . . . . . . . . 11
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑘 ∈ 𝐾) → (◡𝐹 “ 𝑘) ∈ 𝐽) | 
| 34 | 31, 32, 33 | syl2anc 584 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (◡𝐹 “ 𝑘) ∈ 𝐽) | 
| 35 | 8 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (topGen‘𝑏) = 𝐽) | 
| 36 | 34, 35 | eleqtrrd 2843 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (◡𝐹 “ 𝑘) ∈ (topGen‘𝑏)) | 
| 37 |  | simprrl 780 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ ∪ 𝐽) | 
| 38 |  | simprrr 781 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) = 𝑧) | 
| 39 |  | simprlr 779 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑧 ∈ 𝑘) | 
| 40 | 38, 39 | eqeltrd 2840 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝐹‘𝑡) ∈ 𝑘) | 
| 41 | 26 | ffnd 6736 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → 𝐹 Fn ∪ 𝐽) | 
| 42 | 41 | adantrr 717 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝐹 Fn ∪ 𝐽) | 
| 43 |  | elpreima 7077 | . . . . . . . . . . 11
⊢ (𝐹 Fn ∪
𝐽 → (𝑡 ∈ (◡𝐹 “ 𝑘) ↔ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) ∈ 𝑘))) | 
| 44 | 42, 43 | syl 17 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → (𝑡 ∈ (◡𝐹 “ 𝑘) ↔ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) ∈ 𝑘))) | 
| 45 | 37, 40, 44 | mpbir2and 713 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → 𝑡 ∈ (◡𝐹 “ 𝑘)) | 
| 46 |  | tg2 22973 | . . . . . . . . 9
⊢ (((◡𝐹 “ 𝑘) ∈ (topGen‘𝑏) ∧ 𝑡 ∈ (◡𝐹 “ 𝑘)) → ∃𝑚 ∈ 𝑏 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘))) | 
| 47 | 36, 45, 46 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → ∃𝑚 ∈ 𝑏 (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘))) | 
| 48 |  | simprl 770 | . . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝑚 ∈ 𝑏) | 
| 49 |  | eqid 2736 | . . . . . . . . . . 11
⊢ (𝐹 “ 𝑚) = (𝐹 “ 𝑚) | 
| 50 |  | imaeq2 6073 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑚 → (𝐹 “ 𝑥) = (𝐹 “ 𝑚)) | 
| 51 | 50 | rspceeqv 3644 | . . . . . . . . . . 11
⊢ ((𝑚 ∈ 𝑏 ∧ (𝐹 “ 𝑚) = (𝐹 “ 𝑚)) → ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥)) | 
| 52 | 48, 49, 51 | sylancl 586 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥)) | 
| 53 | 42 | adantr 480 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝐹 Fn ∪ 𝐽) | 
| 54 |  | fnfun 6667 | . . . . . . . . . . . . . 14
⊢ (𝐹 Fn ∪
𝐽 → Fun 𝐹) | 
| 55 | 53, 54 | syl 17 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → Fun 𝐹) | 
| 56 |  | simprrr 781 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝑚 ⊆ (◡𝐹 “ 𝑘)) | 
| 57 |  | funimass2 6648 | . . . . . . . . . . . . 13
⊢ ((Fun
𝐹 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)) → (𝐹 “ 𝑚) ⊆ 𝑘) | 
| 58 | 55, 56, 57 | syl2anc 584 | . . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ⊆ 𝑘) | 
| 59 |  | vex 3483 | . . . . . . . . . . . 12
⊢ 𝑘 ∈ V | 
| 60 |  | ssexg 5322 | . . . . . . . . . . . 12
⊢ (((𝐹 “ 𝑚) ⊆ 𝑘 ∧ 𝑘 ∈ V) → (𝐹 “ 𝑚) ∈ V) | 
| 61 | 58, 59, 60 | sylancl 586 | . . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ∈ V) | 
| 62 |  | eqid 2736 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) = (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) | 
| 63 | 62 | elrnmpt 5968 | . . . . . . . . . . 11
⊢ ((𝐹 “ 𝑚) ∈ V → ((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥))) | 
| 64 | 61, 63 | syl 17 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → ((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝑏 (𝐹 “ 𝑚) = (𝐹 “ 𝑥))) | 
| 65 | 52, 64 | mpbird 257 | . . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) | 
| 66 | 38 | adantr 480 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝐹‘𝑡) = 𝑧) | 
| 67 |  | simprrl 780 | . . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝑡 ∈ 𝑚) | 
| 68 |  | cnvimass 6099 | . . . . . . . . . . . . 13
⊢ (◡𝐹 “ 𝑘) ⊆ dom 𝐹 | 
| 69 | 56, 68 | sstrdi 3995 | . . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝑚 ⊆ dom 𝐹) | 
| 70 |  | funfvima2 7252 | . . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ 𝑚 ⊆ dom 𝐹) → (𝑡 ∈ 𝑚 → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚))) | 
| 71 | 55, 69, 70 | syl2anc 584 | . . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝑡 ∈ 𝑚 → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚))) | 
| 72 | 67, 71 | mpd 15 | . . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → (𝐹‘𝑡) ∈ (𝐹 “ 𝑚)) | 
| 73 | 66, 72 | eqeltrrd 2841 | . . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → 𝑧 ∈ (𝐹 “ 𝑚)) | 
| 74 |  | eleq2 2829 | . . . . . . . . . . 11
⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝐹 “ 𝑚))) | 
| 75 |  | sseq1 4008 | . . . . . . . . . . 11
⊢ (𝑤 = (𝐹 “ 𝑚) → (𝑤 ⊆ 𝑘 ↔ (𝐹 “ 𝑚) ⊆ 𝑘)) | 
| 76 | 74, 75 | anbi12d 632 | . . . . . . . . . 10
⊢ (𝑤 = (𝐹 “ 𝑚) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘) ↔ (𝑧 ∈ (𝐹 “ 𝑚) ∧ (𝐹 “ 𝑚) ⊆ 𝑘))) | 
| 77 | 76 | rspcev 3621 | . . . . . . . . 9
⊢ (((𝐹 “ 𝑚) ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∧ (𝑧 ∈ (𝐹 “ 𝑚) ∧ (𝐹 “ 𝑚) ⊆ 𝑘)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 78 | 65, 73, 58, 77 | syl12anc 836 | . . . . . . . 8
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) ∧ (𝑚 ∈ 𝑏 ∧ (𝑡 ∈ 𝑚 ∧ 𝑚 ⊆ (◡𝐹 “ 𝑘)))) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 79 | 47, 78 | rexlimddv 3160 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧))) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 80 | 79 | anassrs 467 | . . . . . 6
⊢
(((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧
(topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) ∧ (𝑡 ∈ ∪ 𝐽 ∧ (𝐹‘𝑡) = 𝑧)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 81 | 30, 80 | rexlimddv 3160 | . . . . 5
⊢ ((((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝑘)) → ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 82 | 81 | ralrimivva 3201 | . . . 4
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∀𝑘 ∈ 𝐾 ∀𝑧 ∈ 𝑘 ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) | 
| 83 |  | basgen2 22997 | . . . 4
⊢ ((𝐾 ∈ Top ∧ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ⊆ 𝐾 ∧ ∀𝑘 ∈ 𝐾 ∀𝑧 ∈ 𝑘 ∃𝑤 ∈ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑘)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) = 𝐾) | 
| 84 | 4, 14, 82, 83 | syl3anc 1372 | . . 3
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) = 𝐾) | 
| 85 | 84, 4 | eqeltrd 2840 | . . . . 5
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top) | 
| 86 |  | tgclb 22978 | . . . . 5
⊢ (ran
(𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases ↔ (topGen‘ran
(𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈ Top) | 
| 87 | 85, 86 | sylibr 234 | . . . 4
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases) | 
| 88 |  | omelon 9687 | . . . . . . 7
⊢ ω
∈ On | 
| 89 |  | simprl 770 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ≼ ω) | 
| 90 |  | ondomen 10078 | . . . . . . 7
⊢ ((ω
∈ On ∧ 𝑏 ≼
ω) → 𝑏 ∈
dom card) | 
| 91 | 88, 89, 90 | sylancr 587 | . . . . . 6
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝑏 ∈ dom card) | 
| 92 | 13 | ffnd 6736 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏) | 
| 93 |  | dffn4 6825 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) Fn 𝑏 ↔ (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) | 
| 94 | 92, 93 | sylib 218 | . . . . . 6
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) | 
| 95 |  | fodomnum 10098 | . . . . . 6
⊢ (𝑏 ∈ dom card → ((𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)):𝑏–onto→ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏)) | 
| 96 | 91, 94, 95 | sylc 65 | . . . . 5
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏) | 
| 97 |  | domtr 9048 | . . . . 5
⊢ ((ran
(𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ 𝑏 ∧ 𝑏 ≼ ω) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω) | 
| 98 | 96, 89, 97 | syl2anc 584 | . . . 4
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω) | 
| 99 |  | 2ndci 23457 | . . . 4
⊢ ((ran
(𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ∈ TopBases ∧ ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥)) ≼ ω) → (topGen‘ran
(𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈
2ndω) | 
| 100 | 87, 98, 99 | syl2anc 584 | . . 3
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → (topGen‘ran (𝑥 ∈ 𝑏 ↦ (𝐹 “ 𝑥))) ∈
2ndω) | 
| 101 | 84, 100 | eqeltrrd 2841 | . 2
⊢ (((𝜑 ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → 𝐾 ∈
2ndω) | 
| 102 |  | 2ndcomap.3 | . . 3
⊢ (𝜑 → 𝐽 ∈
2ndω) | 
| 103 |  | is2ndc 23455 | . . 3
⊢ (𝐽 ∈ 2ndω
↔ ∃𝑏 ∈
TopBases (𝑏 ≼ ω
∧ (topGen‘𝑏) =
𝐽)) | 
| 104 | 102, 103 | sylib 218 | . 2
⊢ (𝜑 → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) | 
| 105 | 101, 104 | r19.29a 3161 | 1
⊢ (𝜑 → 𝐾 ∈
2ndω) |