Theorem coexd 7924

Description: The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.)

Hypotheses

Ref	Expression
coexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
coexd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
coexd	⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)

Proof of Theorem coexd

Step	Hyp	Ref	Expression
1		coexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		coexd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		coexg 7922	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463   ∘ ccom 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670

This theorem is referenced by:  rhmmpl  22505  rhmply1vr1  22509  rhmply1vsca  22510  1arithidom  33768  esplyfval  33894  aks5lem2  42839  rhmpsr  43200  fuco112  49985  fuco111  49986  fuco112x  49988  prcof21a  50047