Theorem invisoinvl 17751

Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17729 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 2737	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		eqid 2737	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
10	1, 9, 3, 5	idiso 17749	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
11	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶))
12	11	oveqd 7378	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
13	10, 12	eleqtrrd 2840	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
14	1, 2, 3, 4, 5, 6, 7, 8, 5, 13	invco 17732	. . 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 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
16	1, 15, 6, 3, 4, 5	isohom 17737	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
17	16, 7	sseldd 3923	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 15, 9, 3, 4, 8, 5, 17	catlid 17643	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
19	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶))
20	19	oveqd 7378	. . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
21	20	fveq1d 6837	. . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
22	1, 9, 3, 5	idinv 17750	. . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
23	21, 22	eqtrd 2772	. . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
24	23	oveq2d 7377	. . . 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 17737	. . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
26	1, 2, 3, 4, 5, 6	invf 17729	. . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
27	26, 7	ffvelcdmd 7032	. . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
28	25, 27	sseldd 3923	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
29	1, 15, 9, 3, 5, 8, 4, 28	catrid 17644	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
30	24, 29	eqtrd 2772	. . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
31	14, 18, 30	3brtr3d 5117	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
32	1, 2, 3, 5, 4	invsym 17723	. 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 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  Hom chom 17225  compcco 17226  Catccat 17624  Idccid 17625  Invcinv 17706  Isociso 17707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-cat 17628  df-cid 17629  df-sect 17708  df-inv 17709  df-iso 17710

This theorem is referenced by:  invisoinvr  17752  isocoinvid  17754