Theorem isrngoiso 38233

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 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 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 38232	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
6	5	eleq2d 2823	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}))
7		f1oeq1 6770	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→𝑌))
8	7	elrab 3648	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌} ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌))
9	6, 8	bitrdi 287	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  ran crn 5633  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  RingOpscrngo 38149   RingOpsHom crngohom 38215   RingOpsIso crngoiso 38216

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-rngoiso 38231

This theorem is referenced by:  rngoiso1o  38234  rngoisohom  38235  rngoisocnv  38236  rngoisoco  38237