Theorem up1st2nd 49544

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 17798	. . . 4 ⊢ Rel (𝐷 Func 𝐸)
2		up1st2nd.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
3		df-br 5101	. . . . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	5	uprcl 49543	. . . . . 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 7993	. . . 4 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
10	1, 8, 9	sylancr 588	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
11	10	oveq1d 7383	. 2 ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊))
12	11, 2	breqdi 5115	1 ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148   Func cfunc 17790   UP cup 49532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-func 17794  df-up 49533

This theorem is referenced by:  up1st2ndb  49546  uobrcl  49552  uptrar  49575  uptrai  49576  isinito2  49858  isinito3  49859  lanrcl4  49993  lanrcl5  49994  islmd  50024  iscmd  50025  lmddu  50026  cmddu  50027  lmdran  50030  cmdlan  50031