# Metamath Proof Explorer

## Theorem funcringcsetcALTV2lem5

Description: Lemma 5 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV2.r 𝑅 = ( RingCat ‘ 𝑈 )
funcringcsetcALTV2.s 𝑆 = ( SetCat ‘ 𝑈 )
funcringcsetcALTV2.b 𝐵 = ( Base ‘ 𝑅 )
funcringcsetcALTV2.c 𝐶 = ( Base ‘ 𝑆 )
funcringcsetcALTV2.u ( 𝜑𝑈 ∈ WUni )
funcringcsetcALTV2.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcringcsetcALTV2.g ( 𝜑𝐺 = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion funcringcsetcALTV2lem5 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )

### Proof

Step Hyp Ref Expression
1 funcringcsetcALTV2.r 𝑅 = ( RingCat ‘ 𝑈 )
2 funcringcsetcALTV2.s 𝑆 = ( SetCat ‘ 𝑈 )
3 funcringcsetcALTV2.b 𝐵 = ( Base ‘ 𝑅 )
4 funcringcsetcALTV2.c 𝐶 = ( Base ‘ 𝑆 )
5 funcringcsetcALTV2.u ( 𝜑𝑈 ∈ WUni )
6 funcringcsetcALTV2.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcringcsetcALTV2.g ( 𝜑𝐺 = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8 7 adantr ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → 𝐺 = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
9 oveq12 ( ( 𝑥 = 𝑋𝑦 = 𝑌 ) → ( 𝑥 RingHom 𝑦 ) = ( 𝑋 RingHom 𝑌 ) )
10 9 adantl ( ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) ∧ ( 𝑥 = 𝑋𝑦 = 𝑌 ) ) → ( 𝑥 RingHom 𝑦 ) = ( 𝑋 RingHom 𝑌 ) )
11 10 reseq2d ( ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) ∧ ( 𝑥 = 𝑋𝑦 = 𝑌 ) ) → ( I ↾ ( 𝑥 RingHom 𝑦 ) ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
12 simprl ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → 𝑋𝐵 )
13 simprr ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → 𝑌𝐵 )
14 ovexd ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 RingHom 𝑌 ) ∈ V )
15 14 resiexd ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ∈ V )
16 8 11 12 13 15 ovmpod ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )