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| Mirrors > Home > MPE Home > Th. List > invffval | Structured version Visualization version GIF version | ||
| Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| invfval.b | ⊢ 𝐵 = (Base‘𝐶) |
| invfval.n | ⊢ 𝑁 = (Inv‘𝐶) |
| invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| invfval.s | ⊢ 𝑆 = (Sect‘𝐶) |
| Ref | Expression |
|---|---|
| invffval | ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | invfval.n | . 2 ⊢ 𝑁 = (Inv‘𝐶) | |
| 2 | invfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
| 4 | invfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
| 6 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶)) | |
| 7 | invfval.s | . . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆) |
| 9 | 8 | oveqd 7448 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦)) |
| 10 | 8 | oveqd 7448 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥)) |
| 11 | 10 | cnveqd 5886 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦𝑆𝑥)) |
| 12 | 9, 11 | ineq12d 4221 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) |
| 13 | 5, 5, 12 | mpoeq123dv 7508 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
| 14 | df-inv 17792 | . . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) | |
| 15 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 16 | 15, 15 | mpoex 8104 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) ∈ V |
| 17 | 13, 14, 16 | fvmpt 7016 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
| 18 | 2, 17 | syl 17 | . 2 ⊢ (𝜑 → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
| 19 | 1, 18 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ◡ccnv 5684 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 Catccat 17707 Sectcsect 17788 Invcinv 17789 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-inv 17792 |
| This theorem is referenced by: invfval 17803 isoval 17809 |
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