up1st2ndr - Mathbox for Zhi Wang

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Theorem up1st2ndr 48969

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

**Hypotheses**
Ref	Expression
up1st2ndr.1	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
up1st2ndr.2	⊢ (𝜑 → 𝑋(⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷UP𝐸)𝑊)𝑀)

**Assertion**
Ref	Expression
up1st2ndr	⊢ (𝜑 → 𝑋(𝐹(𝐷UP𝐸)𝑊)𝑀)

**Proof of Theorem up1st2ndr**
Step	Hyp	Ref	Expression
1		relfunc 17879	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2ndr.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
3		1st2nd 8046	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5	4	oveq1d 7428	. . 3 ⊢ (𝜑 → (𝐹(𝐷UP𝐸)𝑊) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷UP𝐸)𝑊))
6	5	eqcomd 2740	. 2 ⊢ (𝜑 → (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷UP𝐸)𝑊) = (𝐹(𝐷UP𝐸)𝑊))
7		up1st2ndr.2	. 2 ⊢ (𝜑 → 𝑋(⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷UP𝐸)𝑊)𝑀)
8	6, 7	breqdi 5138	1 ⊢ (𝜑 → 𝑋(𝐹(𝐷UP𝐸)𝑊)𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⟨cop 4612 class class class wbr 5123 Rel wrel 5670 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 2^nd c2nd 7995 Func cfunc 17871 UPcup 48957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-func 17875

This theorem is referenced by: up1st2ndb 48970

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