Metamath Proof Explorer

Theorem funcringcsetcALTV2lem1

Description: Lemma 1 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV2.r 𝑅 = ( RingCat ‘ 𝑈 )
funcringcsetcALTV2.s 𝑆 = ( SetCat ‘ 𝑈 )
funcringcsetcALTV2.b 𝐵 = ( Base ‘ 𝑅 )
funcringcsetcALTV2.c 𝐶 = ( Base ‘ 𝑆 )
funcringcsetcALTV2.u ( 𝜑𝑈 ∈ WUni )
funcringcsetcALTV2.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
Assertion funcringcsetcALTV2lem1 ( ( 𝜑𝑋𝐵 ) → ( 𝐹𝑋 ) = ( Base ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV2.r 𝑅 = ( RingCat ‘ 𝑈 )
2 funcringcsetcALTV2.s 𝑆 = ( SetCat ‘ 𝑈 )
3 funcringcsetcALTV2.b 𝐵 = ( Base ‘ 𝑅 )
4 funcringcsetcALTV2.c 𝐶 = ( Base ‘ 𝑆 )
5 funcringcsetcALTV2.u ( 𝜑𝑈 ∈ WUni )
6 funcringcsetcALTV2.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 6 adantr ( ( 𝜑𝑋𝐵 ) → 𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
8 fveq2 ( 𝑥 = 𝑋 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
9 8 adantl ( ( ( 𝜑𝑋𝐵 ) ∧ 𝑥 = 𝑋 ) → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑋 ) )
10 simpr ( ( 𝜑𝑋𝐵 ) → 𝑋𝐵 )
11 fvexd ( ( 𝜑𝑋𝐵 ) → ( Base ‘ 𝑋 ) ∈ V )
12 7 9 10 11 fvmptd ( ( 𝜑𝑋𝐵 ) → ( 𝐹𝑋 ) = ( Base ‘ 𝑋 ) )