Theorem up1st2nd 49147

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 17800	. . . 4 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2nd.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
3		df-br 5103	. . . . . . 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 2729	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	5	uprcl 49146	. . . . . 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 7997	. . . 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 7384	. 2 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
12	11, 2	breqdi 5117	1 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4591   class class class wbr 5102  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155   Func cfunc 17792   UP cup 49135

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-func 17796  df-up 49136

This theorem is referenced by:  up1st2ndb  49149  uobrcl  49155  uptrar  49178  uptrai  49179  isinito2  49461  isinito3  49462  lanrcl4  49596  lanrcl5  49597  islmd  49627  iscmd  49628  lmddu  49629  cmddu  49630  lmdran  49633  cmdlan  49634