Theorem invcoisoid 17976

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 17975	. . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
9		eqid 2760	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 3, 4, 5, 6, 9	isinv 17944	. . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		simpl 488	. . . 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 2760	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2760	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
16		invcoisoid.1	. . . 4 ⊢ 1 = (Id‘𝐶)
17	1, 14, 2, 4, 5, 6	isohom 17960	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
18	17, 7	sseldd 3910	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
19	1, 14, 2, 4, 6, 5	isohom 17960	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
20	1, 3, 4, 5, 6, 2	invf 17952	. . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
21	20, 7	ffvelcdmd 7090	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
22	19, 21	sseldd 3910	. . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
23	1, 14, 15, 16, 9, 4, 5, 6, 18, 22	issect2 17938	. . 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 2766	. . . . 5 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
27	26	oveqd 7441	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹))
28	27	eqeq1d 2762	. . 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ⟨cop 4834   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  Hom chom 17448  compcco 17449  Catccat 17847  Idccid 17848  Sectcsect 17928  Invcinv 17929  Isociso 17930

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-cat 17851  df-cid 17852  df-sect 17931  df-inv 17932  df-iso 17933

This theorem is used by:  rcaninv  17978  upeu2lem  50250