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Metamath Proof Explorer

Theorem funcestrcsetclem4

Description: Lemma 4 for funcestrcsetc . (Contributed by AV, 22-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 ) ) ) ) )
Assertion funcestrcsetclem4 ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )

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 eqid ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
9 ovex ⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V
10 resiexg ⊢ ( ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V → ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V )
11 9 10 ax-mp ⊢ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ∈ V
12 8 11 fnmpoi ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) Fn ( 𝐵 × 𝐵 )
13 7 fneq1d ⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) Fn ( 𝐵 × 𝐵 ) ) )
14 12 13 mpbiri ⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) )