Theorem funcsetcestrclem2

Description: Lemma 2 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 )
Assertion	funcsetcestrclem2	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )

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	1 2 3	funcsetcestrclem1	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐹 ‘ 𝑋 ) = { ⟨ ( Base ‘ ndx ) , 𝑋 ⟩ } )
7	1 2 4 5	setc1strwun	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → { ⟨ ( Base ‘ ndx ) , 𝑋 ⟩ } ∈ 𝑈 )
8	6 7	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐶 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑈 )