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Theorem invsym2 16344
Description: The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
invsym2 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))

Proof of Theorem invsym2
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . . . 5 𝐵 = (Base‘𝐶)
2 invfval.n . . . . 5 𝑁 = (Inv‘𝐶)
3 invfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
4 invfval.y . . . . 5 (𝜑𝑌𝐵)
5 invfval.x . . . . 5 (𝜑𝑋𝐵)
6 eqid 2621 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 16342 . . . 4 (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8 relxp 5188 . . . 4 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9 relss 5167 . . . 4 ((𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌𝑁𝑋)))
107, 8, 9mpisyl 21 . . 3 (𝜑 → Rel (𝑌𝑁𝑋))
11 relcnv 5462 . . 3 Rel (𝑋𝑁𝑌)
1210, 11jctil 559 . 2 (𝜑 → (Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
131, 2, 3, 5, 4invsym 16343 . . . 4 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔𝑔(𝑌𝑁𝑋)𝑓))
14 vex 3189 . . . . . 6 𝑔 ∈ V
15 vex 3189 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5265 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓𝑓(𝑋𝑁𝑌)𝑔)
17 df-br 4614 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
1816, 17bitr3i 266 . . . 4 (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
19 df-br 4614 . . . 4 (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
2013, 18, 193bitr3g 302 . . 3 (𝜑 → (⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
2120eqrelrdv2 5180 . 2 (((Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → (𝑋𝑁𝑌) = (𝑌𝑁𝑋))
2212, 21mpancom 702 1 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3555  cop 4154   class class class wbr 4613   × cxp 5072  ccnv 5073  Rel wrel 5079  cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  Catccat 16246  Invcinv 16326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-sect 16328  df-inv 16329
This theorem is referenced by:  invf  16349  invf1o  16350  invinv  16351  cicsym  16385
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