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Theorem setciso 17431

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 2758	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2758	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		setcmon.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		setcmon.c	. . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈)
5	4	setccat 17425	. . . . 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 17418	. . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
9	7, 8	eleqtrd 2854	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
10		setcmon.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
11	10, 8	eleqtrd 2854	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		setciso.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
13	1, 2, 6, 9, 11, 12	isoval 17108	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
14	13	eleq2d 2837	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
15	1, 2, 6, 9, 11	invfun 17107	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
16		funfvbrb 6817	. . . . 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 17430	. . . . 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 486	. . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹:𝑋–1-1-onto→𝑌)
20	18, 19	syl6bi 256	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹:𝑋–1-1-onto→𝑌))
21	17, 20	sylbid 243	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹:𝑋–1-1-onto→𝑌))
22		eqid 2758	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
23	4, 3, 7, 10, 2	setcinv 17430	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ^◡𝐹 = ^◡𝐹)))
24		funrel 6357	. . . . . . 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 5790	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
27	26	ex 416	. . . . . 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 263	. . . 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 215	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))
32	14, 31	bitrd 282	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5036 ^◡ccnv 5527 dom cdm 5528 Rel wrel 5533 Fun wfun 6334 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7156 Basecbs 16555 Catccat 17007 Invcinv 17088 Isociso 17089 SetCatcsetc 17415

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-map 8424 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-fz 12953 df-struct 16557 df-ndx 16558 df-slot 16559 df-base 16561 df-hom 16661 df-cco 16662 df-cat 17011 df-cid 17012 df-sect 17090 df-inv 17091 df-iso 17092 df-setc 17416

This theorem is referenced by: yonffthlem 17612

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