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 )