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 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcringcsetcALTV2.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcringcsetcALTV2.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion funcringcsetcALTV2lem6 ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 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 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
6 funcringcsetcALTV2.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcringcsetcALTV2.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8 1 2 3 4 5 6 7 funcringcsetcALTV2lem5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
9 8 3adant3 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑋 RingHom 𝑌 ) ) )
10 9 fveq1d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 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 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( I ↾ ( 𝑋 RingHom 𝑌 ) ) ‘ 𝐻 ) = 𝐻 )
13 10 12 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )