Metamath Proof Explorer


Theorem funcringcsetclem3ALTV

Description: Lemma 3 for funcringcsetcALTV . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV.r 𝑅 = ( RingCatALTV ‘ 𝑈 )
funcringcsetcALTV.s 𝑆 = ( SetCat ‘ 𝑈 )
funcringcsetcALTV.b 𝐵 = ( Base ‘ 𝑅 )
funcringcsetcALTV.c 𝐶 = ( Base ‘ 𝑆 )
funcringcsetcALTV.u ( 𝜑𝑈 ∈ WUni )
funcringcsetcALTV.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
Assertion funcringcsetclem3ALTV ( 𝜑𝐹 : 𝐵𝐶 )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV.r 𝑅 = ( RingCatALTV ‘ 𝑈 )
2 funcringcsetcALTV.s 𝑆 = ( SetCat ‘ 𝑈 )
3 funcringcsetcALTV.b 𝐵 = ( Base ‘ 𝑅 )
4 funcringcsetcALTV.c 𝐶 = ( Base ‘ 𝑆 )
5 funcringcsetcALTV.u ( 𝜑𝑈 ∈ WUni )
6 funcringcsetcALTV.f ( 𝜑𝐹 = ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7 1 3 5 ringcbasbasALTV ( ( 𝜑𝑥𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝑈 )
8 2 5 setcbas ( 𝜑𝑈 = ( Base ‘ 𝑆 ) )
9 8 eqcomd ( 𝜑 → ( Base ‘ 𝑆 ) = 𝑈 )
10 9 adantr ( ( 𝜑𝑥𝐵 ) → ( Base ‘ 𝑆 ) = 𝑈 )
11 7 10 eleqtrrd ( ( 𝜑𝑥𝐵 ) → ( Base ‘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
12 11 4 eleqtrrdi ( ( 𝜑𝑥𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝐶 )
13 12 fmpttd ( 𝜑 → ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵𝐶 )
14 6 feq1d ( 𝜑 → ( 𝐹 : 𝐵𝐶 ↔ ( 𝑥𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵𝐶 ) )
15 13 14 mpbird ( 𝜑𝐹 : 𝐵𝐶 )