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Theorem setciso 18158

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 2740	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2740	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		setcmon.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		setcmon.c	. . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈)
5	4	setccat 18152	. . . . 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 18145	. . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
9	7, 8	eleqtrd 2846	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
10		setcmon.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
11	10, 8	eleqtrd 2846	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		setciso.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
13	1, 2, 6, 9, 11, 12	isoval 17826	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
14	13	eleq2d 2830	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
15	1, 2, 6, 9, 11	invfun 17825	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
16		funfvbrb 7084	. . . . 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 18157	. . . . 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 482	. . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹:𝑋–1-1-onto→𝑌)
20	18, 19	biimtrdi 253	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹:𝑋–1-1-onto→𝑌))
21	17, 20	sylbid 240	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹:𝑋–1-1-onto→𝑌))
22		eqid 2740	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
23	4, 3, 7, 10, 2	setcinv 18157	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ^◡𝐹 = ^◡𝐹)))
24		funrel 6595	. . . . . . 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 5969	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
27	26	ex 412	. . . . . 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 260	. . . 4 ⊢ (𝜑 → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ^◡𝐹 = ^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
30	22, 29	mpan2i 696	. . 3 ⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
31	21, 30	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))
32	14, 31	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 ^◡ccnv 5699 dom cdm 5700 Rel wrel 5705 Fun wfun 6567 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 Catccat 17722 Invcinv 17806 Isociso 17807 SetCatcsetc 18142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-hom 17335 df-cco 17336 df-cat 17726 df-cid 17727 df-sect 17808 df-inv 17809 df-iso 17810 df-setc 18143

This theorem is referenced by: yonffthlem 18352

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