Theorem up1st2ndb 49769

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 488	. . 3 ⊢ ((𝜑 ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
2	1	up1st2nd 49767	. 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 484	. . 3 ⊢ ((𝜑 ∧ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) → 𝐹 ∈ (𝐷 Func 𝐸))
5		simpr 488	. . 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 49768	. 2 ⊢ ((𝜑 ∧ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀)
7	2, 6	impbida 810	1 ⊢ (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  ⟨cop 4585   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391  1^st c1st 7963  2^nd c2nd 7964   Func cfunc 17878   UP cup 49755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-func 17882  df-up 49756

This theorem is referenced by:  uptra  49797  uptr2a  49804  isinito2lem  50080  lanup  50223  iscmd  50248  lmddu  50249