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Theorem invsym2 16246
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 2610 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 16244 . . . 4 (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8 relxp 5150 . . . 4 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9 relss 5129 . . . 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 5422 . . 3 Rel (𝑋𝑁𝑌)
1210, 11jctil 558 . 2 (𝜑 → (Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
131, 2, 3, 5, 4invsym 16245 . . . 4 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔𝑔(𝑌𝑁𝑋)𝑓))
14 vex 3176 . . . . . 6 𝑔 ∈ V
15 vex 3176 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5227 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓𝑓(𝑋𝑁𝑌)𝑔)
17 df-br 4584 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
1816, 17bitr3i 265 . . . 4 (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
19 df-br 4584 . . . 4 (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
2013, 18, 193bitr3g 301 . . 3 (𝜑 → (⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
2120eqrelrdv2 5142 . 2 (((Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → (𝑋𝑁𝑌) = (𝑌𝑁𝑋))
2212, 21mpancom 700 1 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wss 3540  cop 4131   class class class wbr 4583   × cxp 5036  ccnv 5037  Rel wrel 5043  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  Invcinv 16228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-sect 16230  df-inv 16231
This theorem is referenced by:  invf  16251  invf1o  16252  invinv  16253  cicsym  16287
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