rngoiso1o - Mathbox for Jeff Madsen

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Theorem rngoiso1o 36652

Description: A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)

**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
rngoiso1o	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)

**Proof of Theorem rngoiso1o**
Step	Hyp	Ref	Expression
1		rngisoval.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑅)
2		rngisoval.2	. . . 4 ⊢ 𝑋 = ran 𝐺
3		rngisoval.3	. . . 4 ⊢ 𝐽 = (1^st ‘𝑆)
4		rngisoval.4	. . . 4 ⊢ 𝑌 = ran 𝐽
5	1, 2, 3, 4	isrngoiso 36651	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))
6	5	simplbda 500	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
7	6	3impa 1110	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ran crn 5670 –1-1-onto→wf1o 6531 ‘cfv 6532 (class class class)co 7393 1^st c1st 7955 RingOpscrngo 36567 RngHom crnghom 36633 RngIso crngiso 36634

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-rngoiso 36649

This theorem is referenced by: rngoisoco 36655

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