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Metamath Proof Explorer

Theorem funcestrcsetclem3

Description: Lemma 3 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 ) ) )
Assertion funcestrcsetclem3 ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )

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 1 3 5 estrcbasbas ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝑈 )
8 2 5 setcbas ⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝑆 ) )
9 8 eqcomd ⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = 𝑈 )
10 9 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑆 ) = 𝑈 )
11 7 10 eleqtrrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
12 11 4 eleqtrrdi ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝐶 )
13 6 12 fmpt3d ⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )