Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6116 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | |
2 | dmexg 7605 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom 𝐵 ∈ V) | |
3 | rnexg 7606 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
4 | xpexg 7465 | . . 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 5218 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 589 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 Vcvv 3493 ⊆ wss 3934 × cxp 5546 dom cdm 5548 ran crn 5549 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 |
This theorem is referenced by: coex 7627 suppco 7862 supp0cosupp0OLD 7865 imacosuppOLD 7867 fsuppco2 8858 fsuppcor 8859 mapfienlem2 8861 wemapwe 9152 cofsmo 9683 relexpsucnnr 14376 supcvg 15203 imasle 16788 setcco 17335 estrcco 17372 pwsco1mhm 17988 pwsco2mhm 17989 efmndov 18038 efmndcl 18039 symgov 18504 symgcl 18505 gsumval3lem2 19018 gsumzf1o 19024 evls1sca 20478 f1lindf 20958 tngds 23249 climcncf 23500 motplusg 26320 tocycfv 30744 smatfval 31053 eulerpartlemmf 31626 hgt750lemg 31918 cossex 35656 tgrpov 37876 erngmul 37934 erngmul-rN 37942 dvamulr 38140 dvavadd 38143 dvhmulr 38214 mendmulr 39779 relexp0a 40052 choicefi 41453 climexp 41876 dvsinax 42187 stoweidlem27 42303 stoweidlem31 42307 stoweidlem59 42335 uspgrbisymrelALT 44021 rngccoALTV 44250 ringccoALTV 44313 |
Copyright terms: Public domain | W3C validator |