Theorem invf1o 17830

Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism 𝐹 ∈ (𝑋𝐼𝑌) has a unique inverse, denoted by ((Inv‘𝐶)‘𝐹). Remark 3.12 of [Adamek] p. 28. (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)
isoval.n	⊢ 𝐼 = (Iso‘𝐶)

Assertion

Ref	Expression
invf1o	⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))

Proof of Theorem invf1o

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		isoval.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
7	1, 2, 3, 4, 5, 6	invf 17829	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
8	7	ffnd 6748	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
9	1, 2, 3, 5, 4, 6	invf 17829	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌))
10	9	ffnd 6748	. . 3 ⊢ (𝜑 → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
11	1, 2, 3, 4, 5	invsym2 17824	. . . 4 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
12	11	fneq1d 6672	. . 3 ⊢ (𝜑 → (◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋) ↔ (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋)))
13	10, 12	mpbird 257	. 2 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋))
14		dff1o4 6870	. 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋)))
15	8, 13, 14	sylanbrc 582	1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ^◡ccnv 5699   Fn wfn 6568  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Catccat 17722  Invcinv 17806  Isociso 17807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-cat 17726  df-cid 17727  df-sect 17808  df-inv 17809  df-iso 17810

This theorem is referenced by:  invinv  17831