Theorem invf1o 16429

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 16428	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
8		ffn 6045	. . 3 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
10	1, 2, 3, 5, 4, 6	invf 16428	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌))
11		ffn 6045	. . . 4 ⊢ ((𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌) → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))
13	1, 2, 3, 4, 5	invsym2 16423	. . . 4 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
14	13	fneq1d 5981	. . 3 ⊢ (𝜑 → (◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋) ↔ (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋)))
15	12, 14	mpbird 247	. 2 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋))
16		dff1o4 6145	. 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋)))
17	9, 15, 16	sylanbrc 698	1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ^◡ccnv 5113   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Catccat 16325  Invcinv 16405  Isociso 16406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by:  invinv  16430