Theorem coexd 41545

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 7914	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3466   ∘ ccom 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678

This theorem is referenced by:  rhmmpl  41618