rngoisoval - Mathbox for Jeff Madsen

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Theorem rngoisoval 38661

Description: Obsolete theorem, use rimval 20608 instead. The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
rngisoval.1	⊢ 𝐺 = (1^st ‘𝑅)
rngisoval.2	⊢ 𝑋 = ran 𝐺
rngisoval.3	⊢ 𝐽 = (1^st ‘𝑆)
rngisoval.4	⊢ 𝑌 = ran 𝐽

**Assertion**
Ref	Expression
rngoisoval	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Distinct variable groups: 𝑅,𝑓 𝑆,𝑓 𝑓,𝑋 𝑓,𝑌 Allowed substitution hints: 𝐺(𝑓) 𝐽(𝑓)

Proof of Theorem rngoisoval Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 7425	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RingOpsHom 𝑠) = (𝑅 RingOpsHom 𝑆))
2		fveq2 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
3		rngisoval.1	. . . . . . . 8 ⊢ 𝐺 = (1^st ‘𝑅)
4	2, 3	eqtr4di 2818	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
5	4	rneqd 5930	. . . . . 6 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
6		rngisoval.2	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	eqtr4di 2818	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
8	7	f1oeq2d 6820	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠)))
9		fveq2 6885	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = (1^st ‘𝑆))
10		rngisoval.3	. . . . . . . 8 ⊢ 𝐽 = (1^st ‘𝑆)
11	9, 10	eqtr4di 2818	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (1^st ‘𝑠) = 𝐽)
12	11	rneqd 5930	. . . . . 6 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = ran 𝐽)
13		rngisoval.4	. . . . . 6 ⊢ 𝑌 = ran 𝐽
14	12, 13	eqtr4di 2818	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (1^st ‘𝑠) = 𝑌)
15	14	f1oeq3d 6821	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓:𝑋–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
16	8, 15	sylan9bb 519	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠) ↔ 𝑓:𝑋–1-1-onto→𝑌))
17	1, 16	rabeqbidv 3436	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)} = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
18		df-rngoiso 38660	. 2 ⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
19		ovex 7449	. . 3 ⊢ (𝑅 RingOpsHom 𝑆) ∈ V
20	19	rabex 5311	. 2 ⊢ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ∈ V
21	17, 18, 20	ovmpoa 7571	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {crab 3418 ran crn 5664 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 1^st c1st 7986 RingOpscrngo 38578 RingOpsHom crngohom 38644 RingOpsIso crngoiso 38645

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-rngoiso 38660

This theorem is used by: isrngoiso 38662

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