Theorem rimval

Description: The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011) (Revised by AV, 24-Jul-2026)

	Ref	Expression
Hypotheses	rhmf1o.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rhmf1o.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion	rimval	⊢ ( 𝑅 RingIso 𝑆 ) = { 𝑓 ∈ ( 𝑅 RingHom 𝑆 ) ∣ 𝑓 : 𝐵 –1-1-onto→ 𝐶 }

Proof

Step	Hyp	Ref	Expression
1		rhmf1o.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rhmf1o.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
3	1 2	isrim	⊢ ( 𝑥 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝑥 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 : 𝐵 –1-1-onto→ 𝐶 ) )
4		f1oeq1	⊢ ( 𝑓 = 𝑥 → ( 𝑓 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝑥 : 𝐵 –1-1-onto→ 𝐶 ) )
5	4	elrab	⊢ ( 𝑥 ∈ { 𝑓 ∈ ( 𝑅 RingHom 𝑆 ) ∣ 𝑓 : 𝐵 –1-1-onto→ 𝐶 } ↔ ( 𝑥 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 : 𝐵 –1-1-onto→ 𝐶 ) )
6	3 5	bitr4i	⊢ ( 𝑥 ∈ ( 𝑅 RingIso 𝑆 ) ↔ 𝑥 ∈ { 𝑓 ∈ ( 𝑅 RingHom 𝑆 ) ∣ 𝑓 : 𝐵 –1-1-onto→ 𝐶 } )
7	6	eqriv	⊢ ( 𝑅 RingIso 𝑆 ) = { 𝑓 ∈ ( 𝑅 RingHom 𝑆 ) ∣ 𝑓 : 𝐵 –1-1-onto→ 𝐶 }