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Theorem coexd 7953
Description: The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.)
Hypotheses
Ref Expression
coexd.1 (𝜑𝐴𝑉)
coexd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
coexd (𝜑 → (𝐴𝐵) ∈ V)

Proof of Theorem coexd
StepHypRef Expression
1 coexd.1 . 2 (𝜑𝐴𝑉)
2 coexd.2 . 2 (𝜑𝐵𝑊)
3 coexg 7951 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
41, 2, 3syl2anc 584 1 (𝜑 → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480  ccom 5689
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696
This theorem is referenced by:  rhmmpl  22387  rhmply1vr1  22391  rhmply1vsca  22392  1arithidom  33565  aks5lem2  42188  rhmpsr  42562  fuco112  49024  fuco111  49025  fuco112x  49027
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