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Theorem invf 17726

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 17722	. . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
7	6	funfnd 6516	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌))
8		isoval.n	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
9	1, 2, 3, 4, 5, 8	isoval 17723	. . . 4 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
10	9	fneq2d 6579	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ↔ (𝑋𝑁𝑌) Fn dom (𝑋𝑁𝑌)))
11	7, 10	mpbird 258	. 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌))
12		df-rn 5629	. . . 4 ⊢ ran (𝑋𝑁𝑌) = dom ^◡(𝑋𝑁𝑌)
13	1, 2, 3, 4, 5	invsym2 17721	. . . . . 6 ⊢ (𝜑 → ^◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
14	13	dmeqd 5847	. . . . 5 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = dom (𝑌𝑁𝑋))
15	1, 2, 3, 5, 4, 8	isoval 17723	. . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌𝑁𝑋))
16	14, 15	eqtr4d 2777	. . . 4 ⊢ (𝜑 → dom ^◡(𝑋𝑁𝑌) = (𝑌𝐼𝑋))
17	12, 16	eqtrid 2786	. . 3 ⊢ (𝜑 → ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋))
18		eqimss 3973	. . 3 ⊢ (ran (𝑋𝑁𝑌) = (𝑌𝐼𝑋) → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋))
20		df-f 6489	. 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ran (𝑋𝑁𝑌) ⊆ (𝑌𝐼𝑋)))
21	11, 19, 20	sylanbrc 589	1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ⊆ wss 3883 ^◡ccnv 5617 dom cdm 5618 ran crn 5619 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 Catccat 17621 Invcinv 17703 Isociso 17704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-cat 17625 df-cid 17626 df-sect 17705 df-inv 17706 df-iso 17707

This theorem is referenced by: invf1o 17727 invisoinvl 17748 invcoisoid 17750 isocoinvid 17751 rcaninv 17752 ffthiso 17889 initoeu2lem1 17972 upeu2lem 49518 thincciso3 49946

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