Theorem rngimrcl 44162

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

Assertion

Ref	Expression
rngimrcl	⊢ (𝐹 ∈ (𝑅 RngIsom 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-rngisom 44153	. 2 ⊢ RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)})
2	1	elmpocl 7381	1 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  {crab 3142  Vcvv 3494  ^◡ccnv 5548  (class class class)co 7150   RngHomo crngh 44150   RngIsom crngs 44151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-dm 5559  df-iota 6308  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rngisom 44153

This theorem is referenced by:  rngimf1o  44170  rngimrnghm  44171