Theorem isrngoiso 36063

Description: The predicate "is a ring isomorphism between 𝑅 and 𝑆". (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
isrngoiso	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))

Proof of Theorem isrngoiso

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngisoval.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		rngisoval.2	. . . 4 ⊢ 𝑋 = ran 𝐺
3		rngisoval.3	. . . 4 ⊢ 𝐽 = (1st ‘𝑆)
4		rngisoval.4	. . . 4 ⊢ 𝑌 = ran 𝐽
5	1, 2, 3, 4	rngoisoval 36062	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
6	5	eleq2d 2824	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}))
7		f1oeq1 6688	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→𝑌))
8	7	elrab 3617	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌))
9	6, 8	bitrdi 286	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  ran crn 5581  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  RingOpscrngo 35979   RngHom crnghom 36045   RngIso crngiso 36046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-rngoiso 36061

This theorem is referenced by:  rngoiso1o  36064  rngoisohom  36065  rngoisocnv  36066  rngoisoco  36067