This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem funcestrcsetclem5

Description: Lemma 5 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 funcestrcsetclem5 ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )

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 7 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) )
11 fveq2 ⊢ ( 𝑦 = 𝑌 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑌 ) )
12 fveq2 ⊢ ( 𝑥 = 𝑋 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
13 11 12 oveqan12rd ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
14 9 8 oveq12i ⊢ ( 𝑁 ↑m 𝑀 ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) )
15 13 14 eqtr4di ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( 𝑁 ↑m 𝑀 ) )
16 15 reseq2d ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )
17 16 adantl ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )
18 simprl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
19 simprr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
20 ovexd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑁 ↑m 𝑀 ) ∈ V )
21 20 resiexd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( I ↾ ( 𝑁 ↑m 𝑀 ) ) ∈ V )
22 10 17 18 19 21 ovmpod ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑌 ) = ( I ↾ ( 𝑁 ↑m 𝑀 ) ) )