Theorem rngimrcl 20688

Description: Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)

Assertion

Ref	Expression
rngimrcl	⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem rngimrcl

Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rngim 20679	. 2 ⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
2	1	elmpocl 7669	1 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  {crab 3415  Vcvv 3453  ^◡ccnv 5653  (class class class)co 7427   RngHom crnghm 20676   RngIso crngim 20677

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-xp 5660  df-dm 5664  df-iota 6500  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-rngim 20679

This theorem is used by:  rngimf1o  20696  rngimrnghm  20697  rngimcnv  20698