Theorem rngcbas

Description: Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020) (Revised by AV, 8-Mar-2020)

	Ref	Expression
Hypotheses	rngcbas.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
	rngcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	rngcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
Assertion	rngcbas	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbas.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
2		rngcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rngcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		eqidd	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( 𝑈 ∩ Rng ) )
5		eqidd	⊢ ( 𝜑 → ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) = ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) )
6	1 3 4 5	rngcval	⊢ ( 𝜑 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) )
7	6	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) )
8	2	a1i	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
9		eqid	⊢ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) )
10		eqid	⊢ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) )
11		fvexd	⊢ ( 𝜑 → ( ExtStrCat ‘ 𝑈 ) ∈ V )
12	4 5	rnghmresfn	⊢ ( 𝜑 → ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) Fn ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) )
13		inss1	⊢ ( 𝑈 ∩ Rng ) ⊆ 𝑈
14		eqid	⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 )
15	14 3	estrcbas	⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
16	13 15	sseqtrid	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ⊆ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
17	9 10 11 12 16	rescbas	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHom ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) )
18	7 8 17	3eqtr4d	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )