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Metamath Proof Explorer

Theorem funcestrcsetclem1

Description: Lemma 1 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 funcestrcsetclem1 ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )

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 6 adantr ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
8 fveq2 ⊢ ( 𝑥 = 𝑋 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
9 8 adantl ⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = 𝑋 ) → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
10 simpr ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
11 fvexd ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ 𝑋 ) ∈ V )
12 7 9 10 11 fvmptd ⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )