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Theorem setciso 18045

Description: An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

**Hypotheses**
Ref	Expression
setcmon.c	⊢ 𝐶 = (SetCat‘𝑈)
setcmon.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
setcmon.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
setcmon.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
setciso.n	⊢ 𝐼 = (Iso‘𝐶)

**Assertion**
Ref	Expression
setciso	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))

**Proof of Theorem setciso**
Step	Hyp	Ref	Expression
1		eqid 2732	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2732	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		setcmon.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		setcmon.c	. . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈)
5	4	setccat 18039	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		setcmon.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
8	4, 3	setcbas 18032	. . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
9	7, 8	eleqtrd 2835	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
10		setcmon.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
11	10, 8	eleqtrd 2835	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		setciso.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
13	1, 2, 6, 9, 11, 12	isoval 17716	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
14	13	eleq2d 2819	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
15	1, 2, 6, 9, 11	invfun 17715	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
16		funfvbrb 7052	. . . . 5 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
17	15, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
18	4, 3, 7, 10, 2	setcinv 18044	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹)))
19		simpl 483	. . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹:𝑋–1-1-onto→𝑌)
20	18, 19	syl6bi 252	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹:𝑋–1-1-onto→𝑌))
21	17, 20	sylbid 239	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹:𝑋–1-1-onto→𝑌))
22		eqid 2732	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
23	4, 3, 7, 10, 2	setcinv 18044	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ^◡𝐹 = ^◡𝐹)))
24		funrel 6565	. . . . . . 7 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → Rel (𝑋(Inv‘𝐶)𝑌))
25	15, 24	syl 17	. . . . . 6 ⊢ (𝜑 → Rel (𝑋(Inv‘𝐶)𝑌))
26		releldm 5943	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
27	26	ex 413	. . . . . 6 ⊢ (Rel (𝑋(Inv‘𝐶)𝑌) → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
28	25, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
29	23, 28	sylbird 259	. . . 4 ⊢ (𝜑 → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ^◡𝐹 = ^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
30	22, 29	mpan2i 695	. . 3 ⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
31	21, 30	impbid 211	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))
32	14, 31	bitrd 278	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 Rel wrel 5681 Fun wfun 6537 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 Catccat 17612 Invcinv 17696 Isociso 17697 SetCatcsetc 18029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-hom 17225 df-cco 17226 df-cat 17616 df-cid 17617 df-sect 17698 df-inv 17699 df-iso 17700 df-setc 18030

This theorem is referenced by: yonffthlem 18239

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