# Metamath Proof Explorer

## Theorem funcringcsetcALTV2lem6

Description: Lemma 6 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 funcringcsetcALTV2lem6 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )

### 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 1 2 3 4 5 6 7 funcringcsetcALTV2lem5 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
9 8 3adant3 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
10 9 fveq1d ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = ( ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ‘ 𝐻 ) )
11 fvresi ( 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) → ( ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ‘ 𝐻 ) = 𝐻 )
12 11 3ad2ant3 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ‘ 𝐻 ) = 𝐻 )
13 10 12 eqtrd ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )