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Theorem invsym2 17807
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 2737 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 17805 . . . 4 (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8 relxp 5703 . . . 4 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9 relss 5791 . . . 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 6122 . . 3 Rel (𝑋𝑁𝑌)
1210, 11jctil 519 . 2 (𝜑 → (Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
131, 2, 3, 5, 4invsym 17806 . . . 4 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔𝑔(𝑌𝑁𝑋)𝑓))
14 vex 3484 . . . . . 6 𝑔 ∈ V
15 vex 3484 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5893 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓𝑓(𝑋𝑁𝑌)𝑔)
17 df-br 5144 . . . . 5 (𝑔(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
1816, 17bitr3i 277 . . . 4 (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌))
19 df-br 5144 . . . 4 (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
2013, 18, 193bitr3g 313 . . 3 (𝜑 → (⟨𝑔, 𝑓⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
2120eqrelrdv2 5805 . 2 (((Rel (𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → (𝑋𝑁𝑌) = (𝑌𝑁𝑋))
2212, 21mpancom 688 1 (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  cop 4632   class class class wbr 5143   × cxp 5683  ccnv 5684  Rel wrel 5690  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  Invcinv 17789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-sect 17791  df-inv 17792
This theorem is referenced by:  invf  17812  invf1o  17813  invinv  17814  cicsym  17848
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