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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 2724 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invss 16543 | . . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
8 | relxp 5235 | . . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
9 | relss 5315 | . . . 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 5613 | . . 3 ⊢ Rel ◡(𝑋𝑁𝑌) | |
12 | 10, 11 | jctil 561 | . 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋))) |
13 | 1, 2, 3, 5, 4 | invsym 16544 | . . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓)) |
14 | vex 3307 | . . . . . 6 ⊢ 𝑔 ∈ V | |
15 | vex 3307 | . . . . . 6 ⊢ 𝑓 ∈ V | |
16 | 14, 15 | brcnv 5412 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔) |
17 | df-br 4761 | . . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) | |
18 | 16, 17 | bitr3i 266 | . . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ 〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌)) |
19 | df-br 4761 | . . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋)) | |
20 | 13, 18, 19 | 3bitr3g 302 | . . 3 ⊢ (𝜑 → (〈𝑔, 𝑓〉 ∈ ◡(𝑋𝑁𝑌) ↔ 〈𝑔, 𝑓〉 ∈ (𝑌𝑁𝑋))) |
21 | 20 | eqrelrdv2 5328 | . 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
22 | 12, 21 | mpancom 706 | 1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1596 ∈ wcel 2103 ⊆ wss 3680 〈cop 4291 class class class wbr 4760 × cxp 5216 ◡ccnv 5217 Rel wrel 5223 ‘cfv 6001 (class class class)co 6765 Basecbs 15980 Hom chom 16075 Catccat 16447 Invcinv 16527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-ral 3019 df-rex 3020 df-reu 3021 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-id 5128 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-1st 7285 df-2nd 7286 df-sect 16529 df-inv 16530 |
This theorem is referenced by: invf 16550 invf1o 16551 invinv 16552 cicsym 16586 |
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