Theorem isrngoiso 38551

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 38550	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsIso 𝑆) = {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌})
6	5	eleq2d 2855	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RingOpsHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}))
7		f1oeq1 6809	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→𝑌))
8	7	elrab 3659	. 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 290	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  ran crn 5663  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  RingOpscrngo 38467   RingOpsHom crngohom 38533   RingOpsIso crngoiso 38534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rngoiso 38549

This theorem is referenced by:  rngoiso1o  38552  rngoisohom  38553  rngoisocnv  38554  rngoisoco  38555