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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 6656 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | invfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6656 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶)) | |
7 | invfval.s | . . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶) | |
8 | 6, 7 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆) |
9 | 8 | oveqd 7159 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦)) |
10 | 8 | oveqd 7159 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥)) |
11 | 10 | cnveqd 5732 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦𝑆𝑥)) |
12 | 9, 11 | ineq12d 4178 | . . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) |
13 | 5, 5, 12 | mpoeq123dv 7215 | . . . 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 17001 | . . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) | |
15 | 4 | fvexi 6670 | . . . . 5 ⊢ 𝐵 ∈ V |
16 | 15, 15 | mpoex 7763 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) ∈ V |
17 | 13, 14, 16 | fvmpt 6754 | . . 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 2868 | 1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3923 ◡ccnv 5540 ‘cfv 6341 (class class class)co 7142 ∈ cmpo 7144 Basecbs 16466 Catccat 16918 Sectcsect 16997 Invcinv 16998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-inv 17001 |
This theorem is referenced by: invfval 17012 isoval 17018 |
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