Theorem invcoisoid 17845

Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-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))
invcoisoid.1	⊢ 1 = (Id‘𝐶)
invcoisoid.o	⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)

Assertion

Ref	Expression
invcoisoid	⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))

Proof of Theorem invcoisoid

Step	Hyp	Ref	Expression
1		invisoinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		invisoinv.i	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
3		invisoinv.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
4		invisoinv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		invisoinv.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		invisoinv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		invisoinv.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
8	1, 2, 3, 4, 5, 6, 7	invisoinvr 17844	. . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
9		eqid 2769	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 3, 4, 5, 6, 9	isinv 17813	. . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		simpl 487	. . . 4 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
12	10, 11	biimtrdi 256	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)))
13	8, 12	mpd 16	. 2 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
14		eqid 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2769	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
16		invcoisoid.1	. . . 4 ⊢ 1 = (Id‘𝐶)
17	1, 14, 2, 4, 5, 6	isohom 17829	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
18	17, 7	sseldd 3946	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
19	1, 14, 2, 4, 6, 5	isohom 17829	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
20	1, 3, 4, 5, 6, 2	invf 17821	. . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
21	20, 7	ffvelcdmd 7078	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
22	19, 21	sseldd 3946	. . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
23	1, 14, 15, 16, 9, 4, 5, 6, 18, 22	issect2 17807	. . 3 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋)))
24		invcoisoid.o	. . . . . . 7 ⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋))
26	25	eqcomd 2775	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
27	26	oveqd 7425	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹))
28	27	eqeq1d 2771	. . 3 ⊢ (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)))
29	23, 28	bitrd 282	. 2 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)))
30	13, 29	mpbid 235	1 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4597   class class class wbr 5110  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  Hom chom 17317  compcco 17318  Catccat 17716  Idccid 17717  Sectcsect 17797  Invcinv 17798  Isociso 17799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-cat 17720  df-cid 17721  df-sect 17800  df-inv 17801  df-iso 17802

This theorem is referenced by:  rcaninv  17847  upeu2lem  49684