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Theorem ringciso 20782

Description: An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)

**Hypotheses**
Ref	Expression
ringcsect.c	⊢ 𝐶 = (RingCat‘𝑈)
ringcsect.b	⊢ 𝐵 = (Base‘𝐶)
ringcsect.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringcsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringcsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringciso.n	⊢ 𝐼 = (Iso‘𝐶)

**Assertion**
Ref	Expression
ringciso	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

**Proof of Theorem ringciso**
Step	Hyp	Ref	Expression
1		ringcsect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2762	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		ringcsect.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcsect.c	. . . . . 6 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 20773	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		ringcsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		ringcsect.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		ringciso.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
10	1, 2, 6, 7, 8, 9	isoval 17828	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
11	10	eleq2d 2848	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
12	1, 2, 6, 7, 8	invfun 17827	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
13		funfvbrb 7046	. . . . 5 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
14	12, 13	syl 18	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
15	4, 1, 3, 7, 8, 2	ringcinv 20781	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹)))
16		simpl 487	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌))
17	15, 16	biimtrdi 256	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
18	14, 17	sylbid 243	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
19		eqid 2762	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
20	4, 1, 3, 7, 8, 2	ringcinv 20781	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹)))
21		funrel 6553	. . . . . . 7 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → Rel (𝑋(Inv‘𝐶)𝑌))
22	12, 21	syl 18	. . . . . 6 ⊢ (𝜑 → Rel (𝑋(Inv‘𝐶)𝑌))
23		releldm 5933	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
24	23	ex 417	. . . . . 6 ⊢ (Rel (𝑋(Inv‘𝐶)𝑌) → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
25	22, 24	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
26	20, 25	sylbird 263	. . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
27	19, 26	mpan2i 709	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
28	18, 27	impbid 215	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
29	11, 28	bitrd 282	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 class class class wbr 5108 ^◡ccnv 5659 dom cdm 5660 Rel wrel 5665 Fun wfun 6530 ‘cfv 6536 (class class class)co 7412 Basecbs 17275 Catccat 17726 Invcinv 17808 Isociso 17809 RingIso crs 20559 RingCatcringc 20755

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-hom 17340 df-cco 17341 df-0g 17500 df-cat 17730 df-cid 17731 df-homf 17732 df-sect 17810 df-inv 17811 df-iso 17812 df-ssc 17873 df-resc 17874 df-subc 17875 df-estrc 18185 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-grp 19009 df-ghm 19290 df-mgp 20223 df-ur 20270 df-ring 20323 df-rhm 20561 df-rim 20562 df-ringc 20756

This theorem is used by: (None)

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