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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 6225 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | |
2 | dmexg 7841 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom 𝐵 ∈ V) | |
3 | rnexg 7842 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
4 | xpexg 7685 | . . 3 ⊢ ((dom 𝐵 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐵 × ran 𝐴) ∈ V) | |
5 | 2, 3, 4 | syl2anr 598 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (dom 𝐵 × ran 𝐴) ∈ V) |
6 | ssexg 5281 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 588 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 × cxp 5632 dom cdm 5634 ran crn 5635 ∘ ccom 5638 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 |
This theorem is referenced by: coex 7868 suppco 8138 fsuppco2 9344 fsuppcor 9345 mapfienlem2 9347 wemapwe 9638 cofsmo 10210 relexpsucnnr 14916 supcvg 15746 imasle 17410 setcco 17974 estrcco 18022 pwsco1mhm 18647 pwsco2mhm 18648 efmndov 18696 efmndcl 18697 symgov 19170 symgcl 19171 gsumval3lem2 19688 gsumzf1o 19694 f1lindf 21244 evls1sca 21705 tngds 24027 tngdsOLD 24028 climcncf 24279 motplusg 27526 tocycfv 32007 smatfval 32433 eulerpartlemmf 33032 hgt750lemg 33324 cossex 36927 tgrpov 39257 erngmul 39315 erngmul-rN 39323 dvamulr 39521 dvavadd 39524 dvhmulr 39595 coexd 40699 mendmulr 41558 relexp0a 42076 choicefi 43508 climexp 43932 dvsinax 44240 stoweidlem27 44354 stoweidlem31 44358 stoweidlem59 44386 uspgrbisymrelALT 46143 rngccoALTV 46372 ringccoALTV 46435 itcoval1 46835 itcoval2 46836 itcoval3 46837 itcovalsucov 46840 |
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