Theorem rimrcl 20597

Description: Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)

Assertion

Ref	Expression
rimrcl	⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem rimrcl

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

Step	Hyp	Ref	Expression
1		df-rim 20588	. 2 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
2	1	elmpocl 7664	1 ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  {crab 3419  Vcvv 3458  ^◡ccnv 5665  (class class class)co 7423   RingHom crh 20584   RingIso crs 20585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-dm 5676  df-iota 6499  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-rim 20588

This theorem is used by:  isrim0  20598  rimisrngim  20620