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Mirrors > Home > MPE Home > Th. List > invfval | 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 |
---|---|
invfval | ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . 3 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.s | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
6 | 1, 2, 3, 4, 4, 5 | invffval 17020 | . 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
7 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
8 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
9 | 7, 8 | oveq12d 7166 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌)) |
10 | 8, 7 | oveq12d 7166 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋)) |
11 | 10 | cnveqd 5739 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑦𝑆𝑥) = ◡(𝑌𝑆𝑋)) |
12 | 9, 11 | ineq12d 4188 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
13 | invfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | ovex 7181 | . . . 4 ⊢ (𝑋𝑆𝑌) ∈ V | |
15 | 14 | inex1 5212 | . . 3 ⊢ ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V) |
17 | 6, 12, 4, 13, 16 | ovmpod 7294 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ∩ cin 3933 ◡ccnv 5547 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 Catccat 16927 Sectcsect 17006 Invcinv 17007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-inv 17010 |
This theorem is referenced by: isinv 17022 invss 17023 dfiso2 17034 oppcinv 17042 |
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