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Theorem invf 16629

Description: The inverse relation is a function from isomorphisms to isomorphisms. (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
invf	⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))

**Proof of Theorem invf**
Step	Hyp	Ref	Expression
1		invfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invfval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	invfun 16625	. . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
7		funfn 6079	. . . 4 ⊢ (Fun (𝑋𝑁𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8	6, 7	sylib 208	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
9		isoval.n	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
10	1, 2, 3, 4, 5, 9	isoval 16626	. . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
11	10	fneq2d 6143	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
12	8, 11	mpbird 247	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
13		df-rn 5277	. . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ^◡(𝑋𝑁𝑌)
14	1, 2, 3, 4, 5	invsym2 16624	. . . . . 6 ⊢ (𝜑 → ^◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
15	14	dmeqd 5481	. . . . 5 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
16	1, 2, 3, 5, 4, 9	isoval 16626	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
17	15, 16	eqtr4d 2797	. . . 4 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18	13, 17	syl5eq 2806	. . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
19		eqimss 3798	. . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20	18, 19	syl 17	. 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
21		df-f 6053	. 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
22	12, 20, 21	sylanbrc 701	1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ^◡ccnv 5265 dom cdm 5266 ran crn 5267 Fun wfun 6043 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 Catccat 16526 Invcinv 16606 Isociso 16607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-cat 16530 df-cid 16531 df-sect 16608 df-inv 16609 df-iso 16610

This theorem is referenced by: invf1o 16630 invisoinvl 16651 invcoisoid 16653 isocoinvid 16654 rcaninv 16655 ffthiso 16790 initoeu2lem1 16865

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