| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cvmliftmoi.g | . . . . . . . . . 10
⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) | 
| 2 | 1 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) | 
| 3 | 2 | fveq1d 6907 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑀)‘𝑅) = ((𝐹 ∘ 𝑁)‘𝑅)) | 
| 4 |  | cvmliftmolem.4 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (◡𝑀 “ 𝑊)) | 
| 5 |  | cvmliftmolem.8 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝐼) | 
| 6 | 4, 5 | sseldd 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ (◡𝑀 “ 𝑊)) | 
| 7 |  | cvmliftmoi.m | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) | 
| 8 |  | cvmliftmo.y | . . . . . . . . . . . . . . 15
⊢ 𝑌 = ∪
𝐾 | 
| 9 |  | cvmliftmo.b | . . . . . . . . . . . . . . 15
⊢ 𝐵 = ∪
𝐶 | 
| 10 | 8, 9 | cnf 23255 | . . . . . . . . . . . . . 14
⊢ (𝑀 ∈ (𝐾 Cn 𝐶) → 𝑀:𝑌⟶𝐵) | 
| 11 | 7, 10 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀:𝑌⟶𝐵) | 
| 12 | 11 | ffnd 6736 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 Fn 𝑌) | 
| 13 |  | elpreima 7077 | . . . . . . . . . . . 12
⊢ (𝑀 Fn 𝑌 → (𝑅 ∈ (◡𝑀 “ 𝑊) ↔ (𝑅 ∈ 𝑌 ∧ (𝑀‘𝑅) ∈ 𝑊))) | 
| 14 | 12, 13 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑅 ∈ (◡𝑀 “ 𝑊) ↔ (𝑅 ∈ 𝑌 ∧ (𝑀‘𝑅) ∈ 𝑊))) | 
| 15 | 14 | simprbda 498 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑅 ∈ (◡𝑀 “ 𝑊)) → 𝑅 ∈ 𝑌) | 
| 16 | 6, 15 | syldan 591 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝑌) | 
| 17 |  | fvco3 7007 | . . . . . . . . . 10
⊢ ((𝑀:𝑌⟶𝐵 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅))) | 
| 18 | 11, 17 | sylan 580 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅))) | 
| 19 | 16, 18 | syldan 591 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅))) | 
| 20 |  | cvmliftmoi.n | . . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) | 
| 21 | 8, 9 | cnf 23255 | . . . . . . . . . . 11
⊢ (𝑁 ∈ (𝐾 Cn 𝐶) → 𝑁:𝑌⟶𝐵) | 
| 22 | 20, 21 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝑁:𝑌⟶𝐵) | 
| 23 |  | fvco3 7007 | . . . . . . . . . 10
⊢ ((𝑁:𝑌⟶𝐵 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅))) | 
| 24 | 22, 23 | sylan 580 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅))) | 
| 25 | 16, 24 | syldan 591 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅))) | 
| 26 | 3, 19, 25 | 3eqtr3d 2784 | . . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → (𝐹‘(𝑀‘𝑅)) = (𝐹‘(𝑁‘𝑅))) | 
| 27 | 26 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐹‘(𝑀‘𝑅)) = (𝐹‘(𝑁‘𝑅))) | 
| 28 | 14 | simplbda 499 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑅 ∈ (◡𝑀 “ 𝑊)) → (𝑀‘𝑅) ∈ 𝑊) | 
| 29 | 6, 28 | syldan 591 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝑀‘𝑅) ∈ 𝑊) | 
| 30 | 29 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑅) ∈ 𝑊) | 
| 31 |  | fvres 6924 | . . . . . . 7
⊢ ((𝑀‘𝑅) ∈ 𝑊 → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = (𝐹‘(𝑀‘𝑅))) | 
| 32 | 30, 31 | syl 17 | . . . . . 6
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = (𝐹‘(𝑀‘𝑅))) | 
| 33 | 5 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑅 ∈ 𝐼) | 
| 34 |  | fvres 6924 | . . . . . . . . 9
⊢ (𝑅 ∈ 𝐼 → ((𝑁 ↾ 𝐼)‘𝑅) = (𝑁‘𝑅)) | 
| 35 | 33, 34 | syl 17 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑅) = (𝑁‘𝑅)) | 
| 36 |  | eqid 2736 | . . . . . . . . . . 11
⊢ ∪ (𝐾
↾t 𝐼) =
∪ (𝐾 ↾t 𝐼) | 
| 37 |  | cvmliftmolem.5 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → (𝐾 ↾t 𝐼) ∈ Conn) | 
| 38 | 37 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐾 ↾t 𝐼) ∈ Conn) | 
| 39 | 20 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ (𝐾 Cn 𝐶)) | 
| 40 |  | cnvimass 6099 | . . . . . . . . . . . . . . . . 17
⊢ (◡𝑀 “ 𝑊) ⊆ dom 𝑀 | 
| 41 | 40, 11 | fssdm 6754 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (◡𝑀 “ 𝑊) ⊆ 𝑌) | 
| 42 | 41 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → (◡𝑀 “ 𝑊) ⊆ 𝑌) | 
| 43 | 4, 42 | sstrd 3993 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ 𝑌) | 
| 44 | 8 | cnrest 23294 | . . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ (𝐾 Cn 𝐶) ∧ 𝐼 ⊆ 𝑌) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn 𝐶)) | 
| 45 | 39, 43, 44 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn 𝐶)) | 
| 46 |  | cvmliftmo.f | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | 
| 47 | 46 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝜓) → 𝐹 ∈ (𝐶 CovMap 𝐽)) | 
| 48 |  | cvmtop1 35266 | . . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | 
| 49 | 47, 48 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ Top) | 
| 50 | 9 | toptopon 22924 | . . . . . . . . . . . . . . 15
⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵)) | 
| 51 | 49, 50 | sylib 218 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (TopOn‘𝐵)) | 
| 52 |  | df-ima 5697 | . . . . . . . . . . . . . . 15
⊢ (𝑁 “ 𝐼) = ran (𝑁 ↾ 𝐼) | 
| 53 |  | cvmliftmolem.3 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝑇) | 
| 54 |  | elssuni 4936 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇) | 
| 55 | 53, 54 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ ∪ 𝑇) | 
| 56 |  | cvmliftmolem.2 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝜓) → 𝑇 ∈ (𝑆‘𝑈)) | 
| 57 |  | cvmliftmolem.1 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 =
(◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | 
| 58 | 57 | cvmsuni 35275 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) | 
| 59 | 56, 58 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝜓) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) | 
| 60 | 55, 59 | sseqtrd 4019 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (◡𝐹 “ 𝑈)) | 
| 61 |  | imass2 6119 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑊 ⊆ (◡𝐹 “ 𝑈) → (◡𝑀 “ 𝑊) ⊆ (◡𝑀 “ (◡𝐹 “ 𝑈))) | 
| 62 | 60, 61 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝜓) → (◡𝑀 “ 𝑊) ⊆ (◡𝑀 “ (◡𝐹 “ 𝑈))) | 
| 63 | 4, 62 | sstrd 3993 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (◡𝑀 “ (◡𝐹 “ 𝑈))) | 
| 64 | 2 | cnveqd 5885 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝜓) → ◡(𝐹 ∘ 𝑀) = ◡(𝐹 ∘ 𝑁)) | 
| 65 |  | cnvco 5895 | . . . . . . . . . . . . . . . . . . . 20
⊢ ◡(𝐹 ∘ 𝑀) = (◡𝑀 ∘ ◡𝐹) | 
| 66 |  | cnvco 5895 | . . . . . . . . . . . . . . . . . . . 20
⊢ ◡(𝐹 ∘ 𝑁) = (◡𝑁 ∘ ◡𝐹) | 
| 67 | 64, 65, 66 | 3eqtr3g 2799 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝜓) → (◡𝑀 ∘ ◡𝐹) = (◡𝑁 ∘ ◡𝐹)) | 
| 68 | 67 | imaeq1d 6076 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝜓) → ((◡𝑀 ∘ ◡𝐹) “ 𝑈) = ((◡𝑁 ∘ ◡𝐹) “ 𝑈)) | 
| 69 |  | imaco 6270 | . . . . . . . . . . . . . . . . . 18
⊢ ((◡𝑀 ∘ ◡𝐹) “ 𝑈) = (◡𝑀 “ (◡𝐹 “ 𝑈)) | 
| 70 |  | imaco 6270 | . . . . . . . . . . . . . . . . . 18
⊢ ((◡𝑁 ∘ ◡𝐹) “ 𝑈) = (◡𝑁 “ (◡𝐹 “ 𝑈)) | 
| 71 | 68, 69, 70 | 3eqtr3g 2799 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝜓) → (◡𝑀 “ (◡𝐹 “ 𝑈)) = (◡𝑁 “ (◡𝐹 “ 𝑈))) | 
| 72 | 63, 71 | sseqtrd 4019 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (◡𝑁 “ (◡𝐹 “ 𝑈))) | 
| 73 | 22 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝜓) → 𝑁:𝑌⟶𝐵) | 
| 74 | 73 | ffund 6739 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝜓) → Fun 𝑁) | 
| 75 | 73 | fdmd 6745 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝜓) → dom 𝑁 = 𝑌) | 
| 76 | 43, 75 | sseqtrrd 4020 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ dom 𝑁) | 
| 77 |  | funimass3 7073 | . . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝑁 ∧ 𝐼 ⊆ dom 𝑁) → ((𝑁 “ 𝐼) ⊆ (◡𝐹 “ 𝑈) ↔ 𝐼 ⊆ (◡𝑁 “ (◡𝐹 “ 𝑈)))) | 
| 78 | 74, 76, 77 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝜓) → ((𝑁 “ 𝐼) ⊆ (◡𝐹 “ 𝑈) ↔ 𝐼 ⊆ (◡𝑁 “ (◡𝐹 “ 𝑈)))) | 
| 79 | 72, 78 | mpbird 257 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → (𝑁 “ 𝐼) ⊆ (◡𝐹 “ 𝑈)) | 
| 80 | 52, 79 | eqsstrrid 4022 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → ran (𝑁 ↾ 𝐼) ⊆ (◡𝐹 “ 𝑈)) | 
| 81 |  | cnvimass 6099 | . . . . . . . . . . . . . . 15
⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹 | 
| 82 |  | cvmcn 35268 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽)) | 
| 83 | 46, 82 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽)) | 
| 84 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . 19
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 85 | 9, 84 | cnf 23255 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽) | 
| 86 | 83, 85 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽) | 
| 87 | 86 | fdmd 6745 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝐹 = 𝐵) | 
| 88 | 87 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → dom 𝐹 = 𝐵) | 
| 89 | 81, 88 | sseqtrid 4025 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → (◡𝐹 “ 𝑈) ⊆ 𝐵) | 
| 90 |  | cnrest2 23295 | . . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran (𝑁 ↾ 𝐼) ⊆ (◡𝐹 “ 𝑈) ∧ (◡𝐹 “ 𝑈) ⊆ 𝐵) → ((𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn 𝐶) ↔ (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn (𝐶 ↾t (◡𝐹 “ 𝑈))))) | 
| 91 | 51, 80, 89, 90 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn 𝐶) ↔ (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn (𝐶 ↾t (◡𝐹 “ 𝑈))))) | 
| 92 | 45, 91 | mpbid 232 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn (𝐶 ↾t (◡𝐹 “ 𝑈)))) | 
| 93 | 92 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾t 𝐼) Cn (𝐶 ↾t (◡𝐹 “ 𝑈)))) | 
| 94 |  | dfss2 3968 | . . . . . . . . . . . . . 14
⊢ (𝑊 ⊆ (◡𝐹 “ 𝑈) ↔ (𝑊 ∩ (◡𝐹 “ 𝑈)) = 𝑊) | 
| 95 | 60, 94 | sylib 218 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (𝑊 ∩ (◡𝐹 “ 𝑈)) = 𝑊) | 
| 96 | 9 | topopn 22913 | . . . . . . . . . . . . . . . 16
⊢ (𝐶 ∈ Top → 𝐵 ∈ 𝐶) | 
| 97 | 49, 96 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ 𝐶) | 
| 98 | 97, 89 | ssexd 5323 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → (◡𝐹 “ 𝑈) ∈ V) | 
| 99 | 57 | cvmsss 35273 | . . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) | 
| 100 | 56, 99 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → 𝑇 ⊆ 𝐶) | 
| 101 | 100, 53 | sseldd 3983 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝐶) | 
| 102 |  | elrestr 17474 | . . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ∈ V ∧ 𝑊 ∈ 𝐶) → (𝑊 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))) | 
| 103 | 49, 98, 101, 102 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → (𝑊 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))) | 
| 104 | 95, 103 | eqeltrrd 2841 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))) | 
| 105 | 104 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑊 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))) | 
| 106 | 57 | cvmscld 35279 | . . . . . . . . . . . . 13
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝑊 ∈ 𝑇) → 𝑊 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈)))) | 
| 107 | 47, 56, 53, 106 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈)))) | 
| 108 | 107 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑊 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈)))) | 
| 109 |  | cvmliftmolem.7 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝐼) | 
| 110 |  | cvmliftmo.k | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ Conn) | 
| 111 |  | conntop 23426 | . . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ Conn → 𝐾 ∈ Top) | 
| 112 | 110, 111 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ Top) | 
| 113 | 112 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Top) | 
| 114 | 8 | restuni 23171 | . . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Top ∧ 𝐼 ⊆ 𝑌) → 𝐼 = ∪ (𝐾 ↾t 𝐼)) | 
| 115 | 113, 43, 114 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝜓) → 𝐼 = ∪ (𝐾 ↾t 𝐼)) | 
| 116 | 109, 115 | eleqtrd 2842 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ ∪ (𝐾 ↾t 𝐼)) | 
| 117 | 116 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑄 ∈ ∪ (𝐾 ↾t 𝐼)) | 
| 118 | 109 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑄 ∈ 𝐼) | 
| 119 |  | fvres 6924 | . . . . . . . . . . . . 13
⊢ (𝑄 ∈ 𝐼 → ((𝑁 ↾ 𝐼)‘𝑄) = (𝑁‘𝑄)) | 
| 120 | 118, 119 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑄) = (𝑁‘𝑄)) | 
| 121 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑄) = (𝑁‘𝑄)) | 
| 122 | 4, 109 | sseldd 3983 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ (◡𝑀 “ 𝑊)) | 
| 123 |  | elpreima 7077 | . . . . . . . . . . . . . . . . 17
⊢ (𝑀 Fn 𝑌 → (𝑄 ∈ (◡𝑀 “ 𝑊) ↔ (𝑄 ∈ 𝑌 ∧ (𝑀‘𝑄) ∈ 𝑊))) | 
| 124 | 12, 123 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑄 ∈ (◡𝑀 “ 𝑊) ↔ (𝑄 ∈ 𝑌 ∧ (𝑀‘𝑄) ∈ 𝑊))) | 
| 125 | 124 | simplbda 499 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑄 ∈ (◡𝑀 “ 𝑊)) → (𝑀‘𝑄) ∈ 𝑊) | 
| 126 | 122, 125 | syldan 591 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝜓) → (𝑀‘𝑄) ∈ 𝑊) | 
| 127 | 126 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑄) ∈ 𝑊) | 
| 128 | 121, 127 | eqeltrrd 2841 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁‘𝑄) ∈ 𝑊) | 
| 129 | 120, 128 | eqeltrd 2840 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑄) ∈ 𝑊) | 
| 130 | 36, 38, 93, 105, 108, 117, 129 | conncn 23435 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼):∪ (𝐾 ↾t 𝐼)⟶𝑊) | 
| 131 | 115 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝐼 = ∪ (𝐾 ↾t 𝐼)) | 
| 132 | 131 | feq2d 6721 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼):𝐼⟶𝑊 ↔ (𝑁 ↾ 𝐼):∪ (𝐾 ↾t 𝐼)⟶𝑊)) | 
| 133 | 130, 132 | mpbird 257 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼):𝐼⟶𝑊) | 
| 134 | 133, 33 | ffvelcdmd 7104 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑅) ∈ 𝑊) | 
| 135 | 35, 134 | eqeltrrd 2841 | . . . . . . 7
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁‘𝑅) ∈ 𝑊) | 
| 136 |  | fvres 6924 | . . . . . . 7
⊢ ((𝑁‘𝑅) ∈ 𝑊 → ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) = (𝐹‘(𝑁‘𝑅))) | 
| 137 | 135, 136 | syl 17 | . . . . . 6
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) = (𝐹‘(𝑁‘𝑅))) | 
| 138 | 27, 32, 137 | 3eqtr4d 2786 | . . . . 5
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅))) | 
| 139 | 57 | cvmsf1o 35278 | . . . . . . . . 9
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝑊 ∈ 𝑇) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈) | 
| 140 | 47, 56, 53, 139 | syl3anc 1372 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈) | 
| 141 |  | f1of1 6846 | . . . . . . . 8
⊢ ((𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈 → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈) | 
| 142 | 140, 141 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈) | 
| 143 | 142 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈) | 
| 144 |  | f1fveq 7283 | . . . . . 6
⊢ (((𝐹 ↾ 𝑊):𝑊–1-1→𝑈 ∧ ((𝑀‘𝑅) ∈ 𝑊 ∧ (𝑁‘𝑅) ∈ 𝑊)) → (((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) ↔ (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 145 | 143, 30, 135, 144 | syl12anc 836 | . . . . 5
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) ↔ (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 146 | 138, 145 | mpbid 232 | . . . 4
⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑅) = (𝑁‘𝑅)) | 
| 147 | 146 | ex 412 | . . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝑀‘𝑄) = (𝑁‘𝑄) → (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 148 | 124 | simprbda 498 | . . . . 5
⊢ ((𝜑 ∧ 𝑄 ∈ (◡𝑀 “ 𝑊)) → 𝑄 ∈ 𝑌) | 
| 149 | 122, 148 | syldan 591 | . . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝑌) | 
| 150 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑄 → (𝑀‘𝑥) = (𝑀‘𝑄)) | 
| 151 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑄 → (𝑁‘𝑥) = (𝑁‘𝑄)) | 
| 152 | 150, 151 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = 𝑄 → ((𝑀‘𝑥) = (𝑁‘𝑥) ↔ (𝑀‘𝑄) = (𝑁‘𝑄))) | 
| 153 | 152 | elrab3 3692 | . . . 4
⊢ (𝑄 ∈ 𝑌 → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑄) = (𝑁‘𝑄))) | 
| 154 | 149, 153 | syl 17 | . . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑄) = (𝑁‘𝑄))) | 
| 155 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑅 → (𝑀‘𝑥) = (𝑀‘𝑅)) | 
| 156 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑅 → (𝑁‘𝑥) = (𝑁‘𝑅)) | 
| 157 | 155, 156 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = 𝑅 → ((𝑀‘𝑥) = (𝑁‘𝑥) ↔ (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 158 | 157 | elrab3 3692 | . . . 4
⊢ (𝑅 ∈ 𝑌 → (𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 159 | 16, 158 | syl 17 | . . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑅) = (𝑁‘𝑅))) | 
| 160 | 147, 154,
159 | 3imtr4d 294 | . 2
⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} → 𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})) | 
| 161 | 22 | ffnd 6736 | . . . . 5
⊢ (𝜑 → 𝑁 Fn 𝑌) | 
| 162 |  | fndmin 7064 | . . . . 5
⊢ ((𝑀 Fn 𝑌 ∧ 𝑁 Fn 𝑌) → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}) | 
| 163 | 12, 161, 162 | syl2anc 584 | . . . 4
⊢ (𝜑 → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}) | 
| 164 | 163 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝜓) → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}) | 
| 165 | 164 | eleq2d 2826 | . 2
⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})) | 
| 166 | 164 | eleq2d 2826 | . 2
⊢ ((𝜑 ∧ 𝜓) → (𝑅 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})) | 
| 167 | 160, 165,
166 | 3imtr4d 294 | 1
⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) → 𝑅 ∈ dom (𝑀 ∩ 𝑁))) |