Theorem coexd 41712

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 7931	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3470   ∘ ccom 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683

This theorem is referenced by:  rhmmpl  41780