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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 7914 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ∘ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ∘ ccom 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 |
This theorem is referenced by: domtr 8998 enfixsn 9076 wdomtr 9565 cfcoflem 10262 axcc3 10428 axdc4uzlem 13943 hashfacen 14408 hashfacenOLD 14409 cofu1st 17828 cofu2nd 17830 cofucl 17833 fucid 17919 sursubmefmnd 18772 injsubmefmnd 18773 smndex1mgm 18783 gsumzaddlem 19780 cnfldfun 20940 cnfldfunALT 20941 cnfldfunALTOLD 20942 znle 21071 selvval 21662 evls1fval 21819 evls1val 21820 evl1fval 21828 evl1val 21829 xkococnlem 23144 xkococn 23145 efmndtmd 23586 pserulm 25915 imsval 29915 tocycf 32253 eulerpartgbij 33308 derangenlem 34099 subfacp1lem5 34112 poimirlem9 36434 poimirlem15 36440 poimirlem17 36442 poimirlem20 36445 mbfresfi 36471 tendopl2 39585 erngplus2 39612 erngplus2-rN 39620 dvaplusgv 39818 dvhvaddass 39905 dvhlveclem 39916 diblss 39978 diblsmopel 39979 dicvaddcl 39998 dicvscacl 39999 cdlemn7 40011 dihordlem7 40022 dihopelvalcpre 40056 xihopellsmN 40062 dihopellsm 40063 rabren3dioph 41485 fzisoeu 43944 stirlinglem14 44737 fundcmpsurinjpreimafv 46010 isomushgr 46428 isomgrtr 46441 |
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