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Mirrors > Home > MPE Home > Th. List > invsym2 | Structured version Visualization version GIF version |
Description: The inverse relation is symmetric. (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) |
Ref | Expression |
---|---|
invsym2 | ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . . 5 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | invfval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | eqid 2726 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invss 17772 | . . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
8 | relxp 5692 | . . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
9 | relss 5779 | . . . 4 ⊢ ((𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌𝑁𝑋))) | |
10 | 7, 8, 9 | mpisyl 21 | . . 3 ⊢ (𝜑 → Rel (𝑌𝑁𝑋)) |
11 | relcnv 6106 | . . 3 ⊢ Rel ◡(𝑋𝑁𝑌) | |
12 | 10, 11 | jctil 518 | . 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋))) |
13 | 1, 2, 3, 5, 4 | invsym 17773 | . . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓)) |
14 | vex 3466 | . . . . . 6 ⊢ 𝑔 ∈ V | |
15 | vex 3466 | . . . . . 6 ⊢ 𝑓 ∈ V | |
16 | 14, 15 | brcnv 5881 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔) |
17 | df-br 5146 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) | |
18 | 16, 17 | bitr3i 276 | . . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) |
19 | df-br 5146 | . . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋)) | |
20 | 13, 18, 19 | 3bitr3g 312 | . . 3 ⊢ (𝜑 → (〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌) ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋))) |
21 | 20 | eqrelrdv2 5793 | . 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
22 | 12, 21 | mpancom 686 | 1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 〈cop 4629 class class class wbr 5145 × cxp 5672 ◡ccnv 5673 Rel wrel 5679 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 Hom chom 17272 Catccat 17672 Invcinv 17756 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-sect 17758 df-inv 17759 |
This theorem is referenced by: invf 17779 invf1o 17780 invinv 17781 cicsym 17815 |
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