Theorem up1st2nd 49843

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 17915	. . . 4 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2nd.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
3		df-br 5111	. . . . . . 7 ⊢ (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ ⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊))
4	2, 3	sylib 221	. . . . . 6 ⊢ (𝜑 → ⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊))
5		eqid 2769	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	5	uprcl 49842	. . . . . 6 ⊢ (⟨𝑋, 𝑀⟩ ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ (Base‘𝐸)))
7	4, 6	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ (Base‘𝐸)))
8	7	simpld 499	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
9		1st2nd 8032	. . . 4 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
10	1, 8, 9	sylancr 598	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
11	10	oveq1d 7423	. 2 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
12	11, 2	breqdi 5125	1 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4597   class class class wbr 5110  Rel wrel 5664  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17265   Func cfunc 17907   UP cup 49831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-func 17911  df-up 49832

This theorem is referenced by:  up1st2ndb  49845  uobrcl  49851  uptrar  49874  uptrai  49875  isinito2  50157  isinito3  50158  lanrcl4  50292  lanrcl5  50293  islmd  50323  iscmd  50324  lmddu  50325  cmddu  50326  lmdran  50329  cmdlan  50330