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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 2823 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invss 17033 | . . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
8 | relxp 5575 | . . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
9 | relss 5658 | . . . 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 5969 | . . 3 ⊢ Rel ◡(𝑋𝑁𝑌) | |
12 | 10, 11 | jctil 522 | . 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋))) |
13 | 1, 2, 3, 5, 4 | invsym 17034 | . . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓)) |
14 | vex 3499 | . . . . . 6 ⊢ 𝑔 ∈ V | |
15 | vex 3499 | . . . . . 6 ⊢ 𝑓 ∈ V | |
16 | 14, 15 | brcnv 5755 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔) |
17 | df-br 5069 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) | |
18 | 16, 17 | bitr3i 279 | . . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) |
19 | df-br 5069 | . . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋)) | |
20 | 13, 18, 19 | 3bitr3g 315 | . . 3 ⊢ (𝜑 → (〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌) ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋))) |
21 | 20 | eqrelrdv2 5670 | . 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 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 × cxp 5555 ◡ccnv 5556 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 Catccat 16937 Invcinv 17017 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-sect 17019 df-inv 17020 |
This theorem is referenced by: invf 17040 invf1o 17041 invinv 17042 cicsym 17076 |
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