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| Mirrors > Home > MPE Home > Th. List > coexg | Structured version Visualization version GIF version | ||
| Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
| Ref | Expression |
|---|---|
| coexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cossxp 6292 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | |
| 2 | dmexg 7923 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom 𝐵 ∈ V) | |
| 3 | rnexg 7924 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
| 4 | xpexg 7770 | . . 3 ⊢ ((dom 𝐵 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐵 × ran 𝐴) ∈ V) | |
| 5 | 2, 3, 4 | syl2anr 597 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (dom 𝐵 × ran 𝐴) ∈ V) |
| 6 | ssexg 5323 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
| 7 | 1, 5, 6 | sylancr 587 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ran crn 5686 ∘ 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: coex 7952 coexd 7953 suppco 8231 fsuppco2 9443 fsuppcor 9444 mapfienlem2 9446 wemapwe 9737 cofsmo 10309 relexpsucnnr 15064 supcvg 15892 imasle 17568 setcco 18128 estrcco 18174 pwsco1mhm 18845 pwsco2mhm 18846 efmndov 18894 efmndcl 18895 symgov 19401 symgcl 19402 gsumval3lem2 19924 gsumzf1o 19930 f1lindf 21842 evls1sca 22327 tngds 24668 tngdsOLD 24669 climcncf 24926 motplusg 28550 tocycfv 33129 smatfval 33794 eulerpartlemmf 34377 hgt750lemg 34669 cossex 38420 tgrpov 40750 erngmul 40808 erngmul-rN 40816 dvamulr 41014 dvavadd 41017 dvhmulr 41088 mendmulr 43196 relexp0a 43729 choicefi 45205 climexp 45620 dvsinax 45928 stoweidlem27 46042 stoweidlem31 46046 stoweidlem59 46074 grimco 47880 uspgrbisymrelALT 48071 rngccoALTV 48187 ringccoALTV 48221 itcoval1 48584 itcoval2 48585 itcoval3 48586 itcovalsucov 48589 |
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