Theorem risci 37988

Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
risci	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Proof of Theorem risci

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2806	. . 3 ⊢ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))
2		risc 37987	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
3	1, 2	imbitrrid 246	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → 𝑅 ≃𝑟 𝑆))
4	3	3impia 1117	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109   class class class wbr 5110  (class class class)co 7390  RingOpscrngo 37895   RingOpsIso crngoiso 37962   ≃_𝑟 crisc 37963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-iota 6467  df-fv 6522  df-ov 7393  df-risc 37984

This theorem is referenced by:  riscer  37989