Theorem up1st2ndr

Description: Combine separated parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025)

	Ref	Expression
Hypotheses	up1st2ndr.1	$$⊢\phi \to F\in \left(D\mathrm{Func}E\right)$$
	up1st2ndr.2	No typesetting found for |- ( ph -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) with typecode |-
Assertion	up1st2ndr	Could not format assertion : No typesetting found for |- ( ph -> X ( F ( D UP E ) W ) M ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		up1st2ndr.1	$$⊢\phi \to F\in \left(D\mathrm{Func}E\right)$$
2		up1st2ndr.2	Could not format ( ph -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) : No typesetting found for |- ( ph -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) with typecode |-
3		relfunc	$$⊢\mathrm{Rel}\left(D\mathrm{Func}E\right)$$
4		1st2nd	$$⊢\left(\mathrm{Rel}\left(D\mathrm{Func}E\right)\wedge F\in \left(D\mathrm{Func}E\right)\right)\to F=⟨1^{st}\left(F\right),2^{nd}\left(F\right)⟩$$
5	3 1 4	sylancr	$$⊢\phi \to F=⟨1^{st}\left(F\right),2^{nd}\left(F\right)⟩$$
6	5	oveq1d	Could not format ( ph -> ( F ( D UP E ) W ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) ) : No typesetting found for |- ( ph -> ( F ( D UP E ) W ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) ) with typecode |-
7	6	eqcomd	Could not format ( ph -> ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) = ( F ( D UP E ) W ) ) : No typesetting found for |- ( ph -> ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) = ( F ( D UP E ) W ) ) with typecode |-
8	7 2	breqdi	Could not format ( ph -> X ( F ( D UP E ) W ) M ) : No typesetting found for |- ( ph -> X ( F ( D UP E ) W ) M ) with typecode |-