Theorem invsym2 17811

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	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
invsym2	⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))

Proof of Theorem invsym2

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		invfval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2735	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17809	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8		relxp 5707	. . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9		relss 5794	. . . 4 ⊢ ((𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌𝑁𝑋)))
10	7, 8, 9	mpisyl 21	. . 3 ⊢ (𝜑 → Rel (𝑌𝑁𝑋))
11		relcnv 6125	. . 3 ⊢ Rel ◡(𝑋𝑁𝑌)
12	10, 11	jctil 519	. 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
13	1, 2, 3, 5, 4	invsym 17810	. . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓))
14		vex 3482	. . . . . 6 ⊢ 𝑔 ∈ V
15		vex 3482	. . . . . 6 ⊢ 𝑓 ∈ V
16	14, 15	brcnv 5896	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔)
17		df-br 5149	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
18	16, 17	bitr3i 277	. . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
19		df-br 5149	. . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
20	13, 18, 19	3bitr3g 313	. . 3 ⊢ (𝜑 → (⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
21	20	eqrelrdv2 5808	. 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
22	12, 21	mpancom 688	1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ⟨cop 4637   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  Rel wrel 5694  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  Invcinv 17793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-sect 17795  df-inv 17796

This theorem is referenced by:  invf  17816  invf1o  17817  invinv  17818  cicsym  17852