Theorem up1st2nd

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

	Ref	Expression
Hypothesis	up1st2nd.1	No typesetting found for |- ( ph -> X ( F ( D UP E ) W ) M ) with typecode |-
Assertion	up1st2nd	Could not format assertion : No typesetting found for |- ( ph -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		up1st2nd.1	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 |-
2		relfunc	$$⊢\mathrm{Rel}\left(D\mathrm{Func}E\right)$$
3		df-br	Could not format ( X ( F ( D UP E ) W ) M <-> <. X , M >. e. ( F ( D UP E ) W ) ) : No typesetting found for |- ( X ( F ( D UP E ) W ) M <-> <. X , M >. e. ( F ( D UP E ) W ) ) with typecode |-
4	1 3	sylib	Could not format ( ph -> <. X , M >. e. ( F ( D UP E ) W ) ) : No typesetting found for |- ( ph -> <. X , M >. e. ( F ( D UP E ) W ) ) with typecode |-
5		eqid	$$⊢{\mathrm{Base}}_E={\mathrm{Base}}_E$$
6	5	uprcl	Could not format ( <. X , M >. e. ( F ( D UP E ) W ) -> ( F e. ( D Func E ) /\ W e. ( Base ` E ) ) ) : No typesetting found for |- ( <. X , M >. e. ( F ( D UP E ) W ) -> ( F e. ( D Func E ) /\ W e. ( Base ` E ) ) ) with typecode |-
7	4 6	syl	$$⊢\phi \to \left(F\in \left(D\mathrm{Func}E\right)\wedge W\in {\mathrm{Base}}_E\right)$$
8	7	simpld	$$⊢\phi \to F\in \left(D\mathrm{Func}E\right)$$
9		1st2nd	$$⊢\left(\mathrm{Rel}\left(D\mathrm{Func}E\right)\wedge F\in \left(D\mathrm{Func}E\right)\right)\to F=⟨1^{st}\left(F\right),2^{nd}\left(F\right)⟩$$
10	2 8 9	sylancr	$$⊢\phi \to F=⟨1^{st}\left(F\right),2^{nd}\left(F\right)⟩$$
11	10	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 |-
12	11 1	breqdi	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 |-