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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 6695 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | invfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | eqtr4di 2789 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6695 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶)) | |
7 | invfval.s | . . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶) | |
8 | 6, 7 | eqtr4di 2789 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆) |
9 | 8 | oveqd 7208 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦)) |
10 | 8 | oveqd 7208 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥)) |
11 | 10 | cnveqd 5729 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦𝑆𝑥)) |
12 | 9, 11 | ineq12d 4114 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) |
13 | 5, 5, 12 | mpoeq123dv 7264 | . . . 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 17207 | . . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) | |
15 | 4 | fvexi 6709 | . . . . 5 ⊢ 𝐵 ∈ V |
16 | 15, 15 | mpoex 7828 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) ∈ V |
17 | 13, 14, 16 | fvmpt 6796 | . . 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 | syl5eq 2783 | 1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ◡ccnv 5535 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 Basecbs 16666 Catccat 17121 Sectcsect 17203 Invcinv 17204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-inv 17207 |
This theorem is referenced by: invfval 17218 isoval 17224 |
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