Metamath Proof Explorer

Theorem funcringcsetclem6ALTV

Description: Lemma 6 for funcringcsetcALTV . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV.r ⊢ 𝑅 = ( RingCatALTV ‘ 𝑈 )
funcringcsetcALTV.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
funcringcsetcALTV.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
funcringcsetcALTV.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
funcringcsetcALTV.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcringcsetcALTV.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcringcsetcALTV.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion funcringcsetclem6ALTV ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑋 RingHom 𝑌 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV.r ⊢ 𝑅 = ( RingCatALTV ‘ 𝑈 )
2 funcringcsetcALTV.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3 funcringcsetcALTV.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
4 funcringcsetcALTV.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
5 funcringcsetcALTV.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
6 funcringcsetcALTV.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcringcsetcALTV.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8 1 2 3 4 5 6 7 funcringcsetclem5ALTV ⊢ ( ( 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 )