Theorem up1st2nd 49346

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

Hypothesis

Ref	Expression
up1st2nd.1	⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)

Assertion

Ref	Expression
up1st2nd	⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Proof of Theorem up1st2nd

Step	Hyp	Ref	Expression
1		relfunc 17777	. . . 4 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2nd.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
3		df-br 5096	. . . . . . 7 ⊢ (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ ⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊))
4	2, 3	sylib 218	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊))
5		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	5	uprcl 49345	. . . . . 6 ⊢ (⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ (Base‘𝐸)))
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ (Base‘𝐸)))
8	7	simpld 494	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
9		1st2nd 7980	. . . 4 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
10	1, 8, 9	sylancr 587	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
11	10	oveq1d 7370	. 2 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
12	11, 2	breqdi 5110	1 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   class class class wbr 5095  Rel wrel 5626  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Basecbs 17127   Func cfunc 17769   UP cup 49334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-func 17773  df-up 49335

This theorem is referenced by:  up1st2ndb  49348  uobrcl  49354  uptrar  49377  uptrai  49378  isinito2  49660  isinito3  49661  lanrcl4  49795  lanrcl5  49796  islmd  49826  iscmd  49827  lmddu  49828  cmddu  49829  lmdran  49832  cmdlan  49833