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| 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 2737 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 7 | 1, 2, 3, 4, 5, 6 | invss 17805 | . . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) | 
| 8 | relxp 5703 | . . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
| 9 | relss 5791 | . . . 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 6122 | . . 3 ⊢ Rel ◡(𝑋𝑁𝑌) | |
| 12 | 10, 11 | jctil 519 | . 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋))) | 
| 13 | 1, 2, 3, 5, 4 | invsym 17806 | . . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓)) | 
| 14 | vex 3484 | . . . . . 6 ⊢ 𝑔 ∈ V | |
| 15 | vex 3484 | . . . . . 6 ⊢ 𝑓 ∈ V | |
| 16 | 14, 15 | brcnv 5893 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔) | 
| 17 | df-br 5144 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) | |
| 18 | 16, 17 | bitr3i 277 | . . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) | 
| 19 | df-br 5144 | . . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋)) | |
| 20 | 13, 18, 19 | 3bitr3g 313 | . . 3 ⊢ (𝜑 → (〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌) ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋))) | 
| 21 | 20 | eqrelrdv2 5805 | . 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) | 
| 22 | 12, 21 | mpancom 688 | 1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 × cxp 5683 ◡ccnv 5684 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 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-sect 17791 df-inv 17792 | 
| This theorem is referenced by: invf 17812 invf1o 17813 invinv 17814 cicsym 17848 | 
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