Theorem invsym2 17392

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 2738	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17390	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8		relxp 5598	. . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9		relss 5682	. . . 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 6001	. . 3 ⊢ Rel ◡(𝑋𝑁𝑌)
12	10, 11	jctil 519	. 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
13	1, 2, 3, 5, 4	invsym 17391	. . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓))
14		vex 3426	. . . . . 6 ⊢ 𝑔 ∈ V
15		vex 3426	. . . . . 6 ⊢ 𝑓 ∈ V
16	14, 15	brcnv 5780	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔)
17		df-br 5071	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
18	16, 17	bitr3i 276	. . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
19		df-br 5071	. . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
20	13, 18, 19	3bitr3g 312	. . 3 ⊢ (𝜑 → (⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
21	20	eqrelrdv2 5694	. 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
22	12, 21	mpancom 684	1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Catccat 17290  Invcinv 17374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-sect 17376  df-inv 17377

This theorem is referenced by:  invf  17397  invf1o  17398  invinv  17399  cicsym  17433