ringcisoALTV - Mathbox for Alexander van der Vekens

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Theorem ringcisoALTV 48802

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 2739	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		ringcsectALTV.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringcsectALTV.c	. . . . . 6 ⊢ 𝐶 = (RingCatALTV‘𝑈)
5	4	ringccatALTV 48798	. . . . 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 17723	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
11	10	eleq2d 2825	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
12	1, 2, 6, 7, 8	invfun 17722	. . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌))
13		funfvbrb 6992	. . . . 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 48801	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹)))
16		simpl 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ^◡𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌))
17	15, 16	biimtrdi 254	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
18	14, 17	sylbid 241	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹 ∈ (𝑋 RingIso 𝑌)))
19		eqid 2739	. . . 4 ⊢ ^◡𝐹 = ^◡𝐹
20	4, 1, 3, 7, 8, 2	ringcinvALTV 48801	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹)))
21		funrel 6502	. . . . . . 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 5886	. . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
24	23	ex 413	. . . . . 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 261	. . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 RingIso 𝑌) ∧ ^◡𝐹 = ^◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
27	19, 26	mpan2i 703	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋 RingIso 𝑌) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
28	18, 27	impbid 213	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
29	11, 28	bitrd 280	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 class class class wbr 5072 ^◡ccnv 5617 dom cdm 5618 Rel wrel 5623 Fun wfun 6479 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 Catccat 17621 Invcinv 17703 Isociso 17704 RingIso crs 20441 RingCatALTVcringcALTV 48778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 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 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-hom 17235 df-cco 17236 df-0g 17395 df-cat 17625 df-cid 17626 df-sect 17705 df-inv 17706 df-iso 17707 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-grp 18903 df-ghm 19179 df-mgp 20113 df-ur 20154 df-ring 20207 df-rhm 20443 df-rim 20444 df-ringcALTV 48779

This theorem is referenced by: (None)

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