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Theorem invsym2 16629
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 2770 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 16627 . . . 4 (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8 relxp 5266 . . . 4 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9 relss 5346 . . . 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 5644 . . 3 Rel (𝑋𝑁𝑌)
1210, 11jctil 503 . 2 (𝜑 → (Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
131, 2, 3, 5, 4invsym 16628 . . . 4 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔𝑔(𝑌𝑁𝑋)𝑓))
14 vex 3352 . . . . . 6 𝑔 ∈ V
15 vex 3352 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5443 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓𝑓(𝑋𝑁𝑌)𝑔)
17 df-br 4785 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
1816, 17bitr3i 266 . . . 4 (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
19 df-br 4785 . . . 4 (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
2013, 18, 193bitr3g 302 . . 3 (𝜑 → (⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
2120eqrelrdv2 5359 . 2 (((Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → (𝑋𝑁𝑌) = (𝑌𝑁𝑋))
2212, 21mpancom 660 1 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wss 3721  cop 4320   class class class wbr 4784   × cxp 5247  ccnv 5248  Rel wrel 5254  cfv 6031  (class class class)co 6792  Basecbs 16063  Hom chom 16159  Catccat 16531  Invcinv 16611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-sect 16613  df-inv 16614
This theorem is referenced by:  invf  16634  invf1o  16635  invinv  16636  cicsym  16670
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