Theorem invss 17751

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 2728	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
7	1, 2, 3, 4, 5, 6	invfval 17749	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
8		inss1 4231	. . 3 ⊢ ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) ⊆ (𝑋(Sect‘𝐶)𝑌)
9	7, 8	eqsstrdi 4036	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ (𝑋(Sect‘𝐶)𝑌))
10		invss.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
11		eqid 2728	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		eqid 2728	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
13	1, 10, 11, 12, 6, 3, 4, 5	sectss 17742	. 2 ⊢ (𝜑 → (𝑋(Sect‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
14	9, 13	sstrd 3992	1 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3948   ⊆ wss 3949   × cxp 5680  ^◡ccnv 5681  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  compcco 17252  Catccat 17651  Idccid 17652  Sectcsect 17734  Invcinv 17735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-sect 17737  df-inv 17738

This theorem is referenced by:  invsym2  17753  invfun  17754  isohom  17766  invfuc  17973