Theorem invisoinvl 17759

Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17737 and is therefore denoted by ((𝑋𝑁𝑌)‘𝐹), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.)

Hypotheses

Ref	Expression
invisoinv.b	⊢ 𝐵 = (Base‘𝐶)
invisoinv.i	⊢ 𝐼 = (Iso‘𝐶)
invisoinv.n	⊢ 𝑁 = (Inv‘𝐶)
invisoinv.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invisoinv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invisoinv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
invisoinv.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))

Assertion

Ref	Expression
invisoinvl	⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)

Proof of Theorem invisoinvl

Step	Hyp	Ref	Expression
1		invisoinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		invisoinv.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
3		invisoinv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invisoinv.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invisoinv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		invisoinv.i	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
7		invisoinv.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
8		eqid 2730	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		eqid 2730	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
10	1, 9, 3, 5	idiso 17757	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
11	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶))
12	11	oveqd 7407	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
13	10, 12	eleqtrrd 2832	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
14	1, 2, 3, 4, 5, 6, 7, 8, 5, 13	invco 17740	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹)(𝑋𝑁𝑌)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))))
15		eqid 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
16	1, 15, 6, 3, 4, 5	isohom 17745	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
17	16, 7	sseldd 3950	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 15, 9, 3, 4, 8, 5, 17	catlid 17651	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
19	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶))
20	19	oveqd 7407	. . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
21	20	fveq1d 6863	. . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
22	1, 9, 3, 5	idinv 17758	. . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
23	21, 22	eqtrd 2765	. . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
24	23	oveq2d 7406	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)))
25	1, 15, 6, 3, 5, 4	isohom 17745	. . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
26	1, 2, 3, 4, 5, 6	invf 17737	. . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
27	26, 7	ffvelcdmd 7060	. . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
28	25, 27	sseldd 3950	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
29	1, 15, 9, 3, 5, 8, 4, 28	catrid 17652	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
30	24, 29	eqtrd 2765	. . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
31	14, 18, 30	3brtr3d 5141	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
32	1, 2, 3, 5, 4	invsym 17731	. 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
33	31, 32	mpbird 257	1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633  Invcinv 17714  Isociso 17715

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-cat 17636  df-cid 17637  df-sect 17716  df-inv 17717  df-iso 17718

This theorem is referenced by:  invisoinvr  17760  isocoinvid  17762