Theorem rngimrcl 41687

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 41678	. 2 ⊢ RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)})
2	1	elmpt2cl 6774	1 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900  Vcvv 3173  ^◡ccnv 5037  (class class class)co 6549   RngHomo crngh 41675   RngIsom crngs 41676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rngisom 41678

This theorem is referenced by:  rngimf1o  41695  rngimrnghm  41696