Theorem up1st2nd 50262

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 18030	. . . 4 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2nd.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
3		df-br 5104	. . . . . . 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 2761	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	5	uprcl 50261	. . . . . 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 500	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
9		1st2nd 8048	. . . 4 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
10	1, 8, 9	sylancr 599	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
11	10	oveq1d 7433	. 2 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
12	11, 2	breqdi 5118	1 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  Rel wrel 5656  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380   Func cfunc 18022   UP cup 50250

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-func 18026  df-up 50251

This theorem is used by:  up1st2ndb  50264  uobrcl  50270  uptrar  50293  uptrai  50294  isinito2  50576  isinito3  50577  lanrcl4  50711  lanrcl5  50712  islmd  50742  iscmd  50743  lmddu  50744  cmddu  50745  lmdran  50748  cmdlan  50749