Theorem up1st2ndb 49149

Description: Combine/separate parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypothesis

Ref	Expression
up1st2ndr.1	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
up1st2ndb	⊢ (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀))

Proof of Theorem up1st2ndb

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
2	1	up1st2nd 49147	. 2 ⊢ ((𝜑 ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)
3		up1st2ndr.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) → 𝐹 ∈ (𝐷 Func 𝐸))
5		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)
6	4, 5	up1st2ndr 49148	. 2 ⊢ ((𝜑 ∧ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
7	2, 6	impbida 800	1 ⊢ (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ⟨cop 4591   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946   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:  uptra  49177  uptr2a  49184  isinito2lem  49460  lanup  49603  iscmd  49628  lmddu  49629