Theorem invss 17806

Description: The inverse relation is a relation between morphisms 𝐹:𝑋⟶𝑌 and their inverses 𝐺:𝑌⟶𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
invss.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
invss	⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

Proof of Theorem invss

Step	Hyp	Ref	Expression
1		invfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invfval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		eqid 2736	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
7	1, 2, 3, 4, 5, 6	invfval 17804	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
8		inss1 4236	. . 3 ⊢ ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) ⊆ (𝑋(Sect‘𝐶)𝑌)
9	7, 8	eqsstrdi 4027	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ (𝑋(Sect‘𝐶)𝑌))
10		invss.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
11		eqid 2736	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		eqid 2736	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
13	1, 10, 11, 12, 6, 3, 4, 5	sectss 17797	. 2 ⊢ (𝜑 → (𝑋(Sect‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
14	9, 13	sstrd 3993	1 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∩ cin 3949   ⊆ wss 3950   × cxp 5682  ^◡ccnv 5683  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Sectcsect 17789  Invcinv 17790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-sect 17792  df-inv 17793

This theorem is referenced by:  invsym2  17808  invfun  17809  isohom  17821  invfuc  18023