Theorem coex 2672

Description: The composition of two sets is a set.

Hypotheses

Ref	Expression
coex.1	⊢ A ∈ V
coex.2	⊢ B ∈ V

Assertion

Ref	Expression
coex	⊢ (A ∘ B) ∈ V

Proof of Theorem coex

Step	Hyp	Ref	Expression
1		coex.1	. 2 ⊢ A ∈ V
2		coex.2	. 2 ⊢ B ∈ V
3		coexg 2671	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∘ B) ∈ V)
4	1, 2, 3	mp2an 520	1 ⊢ (A ∘ B) ∈ V

Syntax hints:   ∈ wcel 1092  Vcvv 1348   ∘ ccom 2414

This theorem is referenced by:  mapenlem1 3384  mapenlem2 3385  ac6lem 3575

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429

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