Theorem up1st2ndr 50410

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 18046	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2ndr.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
3		1st2nd 8054	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
5	4	oveq1d 7439	. . 3 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
6	5	eqcomd 2766	. 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 5117	1 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ⟨cop 4834   class class class wbr 5102  Rel wrel 5656  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004   Func cfunc 18038   UP cup 50397

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-func 18042

This theorem is used by:  up1st2ndb  50411