Theorem brric

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019)

	Ref	Expression
Assertion	brric	⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-ric	⊢ ≃𝑟 = ( ◡ RingIso “ ( V ∖ 1o ) )
2		ovex	⊢ ( 𝑟 RingHom 𝑠 ) ∈ V
3		rabexg	⊢ ( ( 𝑟 RingHom 𝑠 ) ∈ V → { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V )
4	2 3	mp1i	⊢ ( ( 𝑟 ∈ V ∧ 𝑠 ∈ V ) → { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V )
5	4	rgen2	⊢ ∀ 𝑟 ∈ V ∀ 𝑠 ∈ V { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V
6		df-rngiso	⊢ RingIso = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } )
7	6	fnmpo	⊢ ( ∀ 𝑟 ∈ V ∀ 𝑠 ∈ V { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V → RingIso Fn ( V × V ) )
8	5 7	ax-mp	⊢ RingIso Fn ( V × V )
9	1 8	brwitnlem	⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )