Theorem isocoinvid 17760

Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-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)
isocoinvid.o	⊢ ⚬ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)

Assertion

Ref	Expression
isocoinvid	⊢ (𝜑 → (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))

Proof of Theorem isocoinvid

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	invisoinvl 17757	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
9		eqid 2736	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 3, 4, 6, 5, 9	isinv 17727	. . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))))
11		simpl 482	. . . 4 ⊢ ((((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
12	10, 11	biimtrdi 253	. . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
13	8, 12	mpd 15	. 2 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
14		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2736	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
16		invcoisoid.1	. . . 4 ⊢ 1 = (Id‘𝐶)
17	1, 14, 2, 4, 6, 5	isohom 17743	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
18	1, 3, 4, 5, 6, 2	invf 17735	. . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
19	18, 7	ffvelcdmd 7037	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
20	17, 19	sseldd 3922	. . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
21	1, 14, 2, 4, 5, 6	isohom 17743	. . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
22	21, 7	sseldd 3922	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
23	1, 14, 15, 16, 9, 4, 6, 5, 20, 22	issect2 17721	. . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌)))
24		isocoinvid.o	. . . . . . 7 ⊢ ⚬ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → ⚬ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌))
26	25	eqcomd 2742	. . . . 5 ⊢ (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = ⚬ )
27	26	oveqd 7384	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)))
28	27	eqeq1d 2738	. . 3 ⊢ (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌) ↔ (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌)))
29	23, 28	bitrd 279	. 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌)))
30	13, 29	mpbid 232	1 ⊢ (𝜑 → (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17630  Idccid 17631  Sectcsect 17711  Invcinv 17712  Isociso 17713

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-cat 17634  df-cid 17635  df-sect 17714  df-inv 17715  df-iso 17716

This theorem is referenced by:  upeu2lem  49503