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Theorem invf 17781

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)
invss.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invss.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		invss.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invss.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	invfun 17777	. . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
7	6	funfnd 6567	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8		isoval.n	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
9	1, 2, 3, 4, 5, 8	isoval 17778	. . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
10	9	fneq2d 6632	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
11	7, 10	mpbird 257	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
12		df-rn 5665	. . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ^◡(𝑋𝑁𝑌)
13	1, 2, 3, 4, 5	invsym2 17776	. . . . . 6 ⊢ (𝜑 → ^◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
14	13	dmeqd 5885	. . . . 5 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
15	1, 2, 3, 5, 4, 8	isoval 17778	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
16	14, 15	eqtr4d 2773	. . . 4 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋))
17	12, 16	eqtrid 2782	. . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18		eqimss 4017	. . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20		df-f 6535	. 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
21	11, 19, 20	sylanbrc 583	1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3926 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 Catccat 17676 Invcinv 17758 Isociso 17759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-cat 17680 df-cid 17681 df-sect 17760 df-inv 17761 df-iso 17762

This theorem is referenced by: invf1o 17782 invisoinvl 17803 invcoisoid 17805 isocoinvid 17806 rcaninv 17807 ffthiso 17944 initoeu2lem1 18027 upeu2lem 48998 thincciso3 49342

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