ringcisoALTV - Mathbox for Alexander van der Vekens

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Theorem ringcisoALTV 48499

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

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

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

**Proof of Theorem ringcisoALTV**
Step	Hyp	Ref	Expression
1		ringcsectALTV.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2734	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		ringcsectALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcsectALTV.c	. . . . . 6 ⊢ 𝐶 = (RingCatALTV‘𝑈)
5	4	ringccatALTV 48495	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		ringcsectALTV.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		ringcsectALTV.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		ringcisoALTV.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
10	1, 2, 6, 7, 8, 9	isoval 17687	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
11	10	eleq2d 2820	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
12	1, 2, 6, 7, 8	invfun 17686	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
13		funfvbrb 6994	. . . . 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 17	. . . 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	ringcinvALTV 48498	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹)))
16		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌))
17	15, 16	biimtrdi 253	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
18	14, 17	sylbid 240	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
19		eqid 2734	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
20	4, 1, 3, 7, 8, 2	ringcinvALTV 48498	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹)))
21		funrel 6507	. . . . . . 7 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → Rel (𝑋(Inv‘𝐶)𝑌))
22	12, 21	syl 17	. . . . . 6 ⊢ (𝜑 → Rel (𝑋(Inv‘𝐶)𝑌))
23		releldm 5891	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
24	23	ex 412	. . . . . 6 ⊢ (Rel (𝑋(Inv‘𝐶)𝑌) → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
25	22, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
26	20, 25	sylbird 260	. . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
27	19, 26	mpan2i 697	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
28	18, 27	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
29	11, 28	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 class class class wbr 5096 ^◡ccnv 5621 dom cdm 5622 Rel wrel 5627 Fun wfun 6484 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 Catccat 17585 Invcinv 17667 Isociso 17668 RingIso crs 20404 RingCatALTVcringcALTV 48475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-hom 17199 df-cco 17200 df-0g 17359 df-cat 17589 df-cid 17590 df-sect 17669 df-inv 17670 df-iso 17671 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-grp 18864 df-ghm 19140 df-mgp 20074 df-ur 20115 df-ring 20168 df-rhm 20406 df-rim 20407 df-ringcALTV 48476

This theorem is referenced by: (None)

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