Theorem invisoinvl 17851

Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17829 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 2740	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		eqid 2740	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
10	1, 9, 3, 5	idiso 17849	. . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
11	6	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶))
12	11	oveqd 7465	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
13	10, 12	eleqtrrd 2847	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
14	1, 2, 3, 4, 5, 6, 7, 8, 5, 13	invco 17832	. . 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 2740	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
16	1, 15, 6, 3, 4, 5	isohom 17837	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
17	16, 7	sseldd 4009	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 15, 9, 3, 4, 8, 5, 17	catlid 17741	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
19	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶))
20	19	oveqd 7465	. . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
21	20	fveq1d 6922	. . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
22	1, 9, 3, 5	idinv 17850	. . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
23	21, 22	eqtrd 2780	. . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
24	23	oveq2d 7464	. . . 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 17837	. . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
26	1, 2, 3, 4, 5, 6	invf 17829	. . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
27	26, 7	ffvelcdmd 7119	. . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
28	25, 27	sseldd 4009	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
29	1, 15, 9, 3, 5, 8, 4, 28	catrid 17742	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
30	24, 29	eqtrd 2780	. . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
31	14, 18, 30	3brtr3d 5197	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
32	1, 2, 3, 5, 4	invsym 17823	. 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 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Hom chom 17322  compcco 17323  Catccat 17722  Idccid 17723  Invcinv 17806  Isociso 17807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-cat 17726  df-cid 17727  df-sect 17808  df-inv 17809  df-iso 17810

This theorem is referenced by:  invisoinvr  17852  isocoinvid  17854