Theorem up1st2ndr 49159

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(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Assertion

Ref	Expression
up1st2ndr	⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)

Proof of Theorem up1st2ndr

Step	Hyp	Ref	Expression
1		relfunc 17830	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2ndr.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
3		1st2nd 8020	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
5	4	oveq1d 7404	. . 3 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
6	5	eqcomd 2736	. 2 ⊢ (𝜑 → (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊) = (𝐹(𝐷 UP 𝐸)𝑊))
7		up1st2ndr.2	. 2 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)
8	6, 7	breqdi 5124	1 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   class class class wbr 5109  Rel wrel 5645  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969   Func cfunc 17822   UP cup 49146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-func 17826

This theorem is referenced by:  up1st2ndb  49160