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Mirrors > Home > MPE Home > Th. List > coex | Structured version Visualization version GIF version |
Description: The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Ref | Expression |
---|---|
coex.1 | ⊢ 𝐴 ∈ V |
coex.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
coex | ⊢ (𝐴 ∘ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | coex.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | coexg 7159 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ∘ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 ∘ ccom 5147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 |
This theorem is referenced by: domtr 8050 enfixsn 8110 wdomtr 8521 cfcoflem 9132 axcc3 9298 axdc4uzlem 12822 hashfacen 13276 cofu1st 16590 cofu2nd 16592 cofucl 16595 fucid 16678 symgplusg 17855 gsumzaddlem 18367 evls1fval 19732 evls1val 19733 evl1fval 19740 evl1val 19741 cnfldfun 19806 cnfldfunALT 19807 znle 19932 xkococnlem 21510 xkococn 21511 symgtgp 21952 pserulm 24221 imsval 27668 eulerpartgbij 30562 derangenlem 31279 subfacp1lem5 31292 poimirlem9 33548 poimirlem15 33554 poimirlem17 33556 poimirlem20 33559 mbfresfi 33586 tendopl2 36382 erngplus2 36409 erngplus2-rN 36417 dvaplusgv 36615 dvhvaddass 36703 dvhlveclem 36714 diblss 36776 diblsmopel 36777 dicvaddcl 36796 dicvscacl 36797 cdlemn7 36809 dihordlem7 36820 dihopelvalcpre 36854 xihopellsmN 36860 dihopellsm 36861 rabren3dioph 37696 fzisoeu 39828 stirlinglem14 40622 |
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