Theorem invsym2 17725

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)
invss.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invss.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		invss.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		invss.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2729	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17723	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
8		relxp 5656	. . . 4 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
9		relss 5744	. . . 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 6075	. . 3 ⊢ Rel ◡(𝑋𝑁𝑌)
12	10, 11	jctil 519	. 2 ⊢ (𝜑 → (Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)))
13	1, 2, 3, 5, 4	invsym 17724	. . . 4 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ 𝑔(𝑌𝑁𝑋)𝑓))
14		vex 3451	. . . . . 6 ⊢ 𝑔 ∈ V
15		vex 3451	. . . . . 6 ⊢ 𝑓 ∈ V
16	14, 15	brcnv 5846	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ 𝑓(𝑋𝑁𝑌)𝑔)
17		df-br 5108	. . . . 5 ⊢ (𝑔◡(𝑋𝑁𝑌)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
18	16, 17	bitr3i 277	. . . 4 ⊢ (𝑓(𝑋𝑁𝑌)𝑔 ↔ ⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌))
19		df-br 5108	. . . 4 ⊢ (𝑔(𝑌𝑁𝑋)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋))
20	13, 18, 19	3bitr3g 313	. . 3 ⊢ (𝜑 → (⟨𝑔, 𝑓⟩ ∈ ◡(𝑋𝑁𝑌) ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑌𝑁𝑋)))
21	20	eqrelrdv2 5758	. 2 ⊢ (((Rel ◡(𝑋𝑁𝑌) ∧ Rel (𝑌𝑁𝑋)) ∧ 𝜑) → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
22	12, 21	mpancom 688	1 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ⟨cop 4595   class class class wbr 5107   × cxp 5636  ^◡ccnv 5637  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  Catccat 17625  Invcinv 17707

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-sect 17709  df-inv 17710

This theorem is referenced by:  invf  17730  invf1o  17731  invinv  17732  cicsym  17766