up1st2ndr

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

	Ref	Expression
Hypotheses	up1st2ndr.1	$⊢ φ \to F \in (D Func E)$
	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 \|-

Step	Hyp	Ref	Expression
1		up1st2ndr.1	$⊢ φ \to F \in (D Func E)$
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	$⊢ Rel (D Func E)$
4		1st2nd	$⊢ (Rel (D Func E) \land F \in (D Func E)) \to F = ⟨1^{st} (F), 2^{nd} (F)⟩$
5	3 1 4	sylancr	$⊢ φ \to F = ⟨1^{st} (F), 2^{nd} (F)⟩$
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 \|-

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