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Metamath Proof Explorer

Theorem funcestrcsetclem6

Description: Lemma 6 for funcestrcsetc . (Contributed by AV, 23-Mar-2020)

Ref Expression
Hypotheses funcestrcsetc.e ⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
funcestrcsetc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
funcestrcsetc.b ⊢ 𝐵 = ( Base ‘ 𝐸 )
funcestrcsetc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
funcestrcsetc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
funcestrcsetc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
funcestrcsetc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
funcestrcsetc.m ⊢ 𝑀 = ( Base ‘ 𝑋 )
funcestrcsetc.n ⊢ 𝑁 = ( Base ‘ 𝑌 )
Assertion funcestrcsetclem6 ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )

Proof

Step Hyp Ref Expression
1 funcestrcsetc.e ⊢ 𝐸 = ( ExtStrCat ‘ 𝑈 )
2 funcestrcsetc.s ⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3 funcestrcsetc.b ⊢ 𝐵 = ( Base ‘ 𝐸 )
4 funcestrcsetc.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
5 funcestrcsetc.u ⊢ ( 𝜑 → 𝑈 ∈ WUni )
6 funcestrcsetc.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 funcestrcsetc.g ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
8 funcestrcsetc.m ⊢ 𝑀 = ( Base ‘ 𝑋 )
9 funcestrcsetc.n ⊢ 𝑁 = ( Base ‘ 𝑌 )
10 1 2 3 4 5 6 7 8 9 funcestrcsetclem5 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )
11 10 3adant3 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )
12 11 fveq1d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = ( ( I ↾ ( 𝑁 ↑m 𝑀 ) ) ‘ 𝐻 ) )
13 fvresi ⊢ ( 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) → ( ( I ↾ ( 𝑁 ↑m 𝑀 ) ) ‘ 𝐻 ) = 𝐻 )
14 13 3ad2ant3 ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) ) → ( ( I ↾ ( 𝑁 ↑m 𝑀 ) ) ‘ 𝐻 ) = 𝐻 )
15 12 14 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐻 ∈ ( 𝑁 ↑m 𝑀 ) ) → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐻 ) = 𝐻 )