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Metamath Proof Explorer

Theorem funcsetcestrclem6

Description: Lemma 6 for funcsetcestrc . (Contributed by AV, 27-Mar-2020)

Ref Expression
Hypotheses funcsetcestrc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
funcsetcestrc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
funcsetcestrc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 ↦ { ⟨ ( Base ‘ ndx ) , 𝑥 ⟩ } ) )
funcsetcestrc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcsetcestrc.o ⊢ ( 𝜑 → ω ∈ 𝑈 )
funcsetcestrc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) )
Assertion funcsetcestrclem6 ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )

Proof

Step Hyp Ref Expression
1 funcsetcestrc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
2 funcsetcestrc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
3 funcsetcestrc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐶 ↦ { ⟨ ( Base ‘ ndx ) , 𝑥 ⟩ } ) )
4 funcsetcestrc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
5 funcsetcestrc.o ⊢ ( 𝜑 → ω ∈ 𝑈 )
6 funcsetcestrc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) )
7 1 2 3 4 5 6 funcsetcestrclem5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
8 7 3adant3 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑌 ↑m 𝑋 ) ) )
9 8 fveq1d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) )
10 fvresi ⊢ ( 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) → ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) = 𝐻 )
11 10 3ad2ant3 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( I ↾ ( 𝑌 ↑m 𝑋 ) ) ‘ 𝐻 ) = 𝐻 )
12 9 11 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ) ∧ 𝐻 ∈ ( 𝑌 ↑m 𝑋 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )