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Metamath Proof Explorer

Theorem funcsetcestrclem4

Description: Lemma 4 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 funcsetcestrclem4 ( 𝜑 → 𝐺 Fn ( 𝐶 × 𝐶 ) )

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 eqid ⊢ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) = ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) )
8 ovex ⊢ ( 𝑦 ↑m 𝑥 ) ∈ V
9 resiexg ⊢ ( ( 𝑦 ↑m 𝑥 ) ∈ V → ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ∈ V )
10 8 9 ax-mp ⊢ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ∈ V
11 7 10 fnmpoi ⊢ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) Fn ( 𝐶 × 𝐶 )
12 6 fneq1d ⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐶 × 𝐶 ) ↔ ( 𝑥 ∈ 𝐶 , 𝑦 ∈ 𝐶 ↦ ( I ↾ ( 𝑦 ↑m 𝑥 ) ) ) Fn ( 𝐶 × 𝐶 ) ) )
13 11 12 mpbiri ⊢ ( 𝜑 → 𝐺 Fn ( 𝐶 × 𝐶 ) )