Theorem up1st2nd2 49911

Description: Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypothesis

Ref	Expression
up1st2nd2.1	⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊))

Assertion

Ref	Expression
up1st2nd2	⊢ (𝜑 → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋))

Proof of Theorem up1st2nd2

Step	Hyp	Ref	Expression
1		relup 49906	. 2 ⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊)
2		up1st2nd2.1	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊))
3		1st2ndbr 8038	. 2 ⊢ ((Rel (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊)) → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋))
4	1, 2, 3	sylancr 598	1 ⊢ (𝜑 → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋))

Syntax hints:   → wi 4   ∈ wcel 2141   class class class wbr 5108  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984   UP cup 49896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  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 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-func 17914  df-up 49897

This theorem is referenced by: (None)