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Theorem invsym2 17029
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 2801 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 17027 . . . 4 (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8 relxp 5541 . . . 4 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9 relss 5624 . . . 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 5938 . . 3 Rel (𝑋𝑁𝑌)
1210, 11jctil 523 . 2 (𝜑 → (Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
131, 2, 3, 5, 4invsym 17028 . . . 4 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔𝑔(𝑌𝑁𝑋)𝑓))
14 vex 3447 . . . . . 6 𝑔 ∈ V
15 vex 3447 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5721 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓𝑓(𝑋𝑁𝑌)𝑔)
17 df-br 5034 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
1816, 17bitr3i 280 . . . 4 (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
19 df-br 5034 . . . 4 (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
2013, 18, 193bitr3g 316 . . 3 (𝜑 → (⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
2120eqrelrdv2 5636 . 2 (((Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → (𝑋𝑁𝑌) = (𝑌𝑁𝑋))
2212, 21mpancom 687 1 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wss 3884  cop 4534   class class class wbr 5033   × cxp 5521  ccnv 5522  Rel wrel 5528  cfv 6328  (class class class)co 7139  Basecbs 16479  Hom chom 16572  Catccat 16931  Invcinv 17011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-sect 17013  df-inv 17014
This theorem is referenced by:  invf  17034  invf1o  17035  invinv  17036  cicsym  17070
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