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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 7951 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ∘ 𝐵) ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: domtr 9047 enfixsn 9121 wdomtr 9615 cfcoflem 10312 axcc3 10478 axdc4uzlem 14024 hashfacen 14493 cofu1st 17928 cofu2nd 17930 cofucl 17933 fucid 18019 sursubmefmnd 18909 injsubmefmnd 18910 smndex1mgm 18920 gsumzaddlem 19939 cnfldfun 21378 cnfldfunALT 21379 cnfldfunOLD 21391 cnfldfunALTOLD 21392 cnfldfunALTOLDOLD 21393 znle 21551 selvval 22139 evls1fval 22323 evls1val 22324 evl1fval 22332 evl1val 22333 xkococnlem 23667 xkococn 23668 efmndtmd 24109 pserulm 26465 imsval 30704 tocycf 33137 eulerpartgbij 34374 derangenlem 35176 subfacp1lem5 35189 poimirlem9 37636 poimirlem15 37642 poimirlem17 37644 poimirlem20 37647 mbfresfi 37673 tendopl2 40779 erngplus2 40806 erngplus2-rN 40814 dvaplusgv 41012 dvhvaddass 41099 dvhlveclem 41110 diblss 41172 diblsmopel 41173 dicvaddcl 41192 dicvscacl 41193 cdlemn7 41205 dihordlem7 41216 dihopelvalcpre 41250 xihopellsmN 41256 dihopellsm 41257 rabren3dioph 42826 fzisoeu 45312 stirlinglem14 46102 fundcmpsurinjpreimafv 47395 grimco 47880 gricushgr 47886 uspgrlim 47959 grlictr 47975 fuco22natlem 49040 |
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