Theorem up1st2nd2 48943

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

Hypothesis

Ref	Expression
up1st2nd2.1	⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷UP𝐸)𝑊))

Assertion

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

Proof of Theorem up1st2nd2

Step	Hyp	Ref	Expression
1		relup 48938	. 2 ⊢ Rel (𝐹(𝐷UP𝐸)𝑊)
2		up1st2nd2.1	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷UP𝐸)𝑊))
3		1st2ndbr 8035	. 2 ⊢ ((Rel (𝐹(𝐷UP𝐸)𝑊) ∧ 𝑋 ∈ (𝐹(𝐷UP𝐸)𝑊)) → (1st ‘𝑋)(𝐹(𝐷UP𝐸)𝑊)(2nd ‘𝑋))
4	1, 2, 3	sylancr 587	1 ⊢ (𝜑 → (1st ‘𝑋)(𝐹(𝐷UP𝐸)𝑊)(2nd ‘𝑋))

Syntax hints:   → wi 4   ∈ wcel 2107   class class class wbr 5116  Rel wrel 5656  ‘cfv 6527  (class class class)co 7399  1^st c1st 7980  2^nd c2nd 7981  UPcup 48929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  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 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-func 17856  df-up 48930

This theorem is referenced by: (None)