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Mirrors > Home > MPE Home > Th. List > fnfun | Structured version Visualization version GIF version |
Description: A function with domain is a function. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
fnfun | ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 5807 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 df-fn 5807 |
This theorem is referenced by: fnrel 5903 funfni 5905 fnco 5913 fnssresb 5917 ffun 5961 f1fun 6016 f1ofun 6052 fnbrfvb 6146 fvelimab 6163 fvun1 6179 elpreima 6245 respreima 6252 fnsnb 6337 fnprb 6377 fntpb 6378 fconst3 6382 fnfvima 6400 fnunirn 6415 nvof1o 6436 f1eqcocnv 6456 fnexALT 7025 curry1 7156 curry2 7159 suppvalfn 7189 suppfnss 7207 fnsuppres 7209 wfrlem2 7302 wfrlem14 7315 tfrlem4 7362 tfrlem5 7363 tfrlem11 7371 tz7.48-2 7424 tz7.49 7427 fndmeng 7919 fnfi 8123 fodomfi 8124 fczfsuppd 8176 marypha2lem4 8227 inf0 8401 r1elss 8552 dfac8alem 8735 alephfp 8814 dfac3 8827 dfac9 8841 dfac12lem1 8848 dfac12lem2 8849 r1om 8949 cfsmolem 8975 alephsing 8981 zorn2lem1 9201 zorn2lem5 9205 zorn2lem6 9206 zorn2lem7 9207 ttukeylem3 9216 ttukeylem6 9219 smobeth 9287 fpwwe2lem9 9339 wunr1om 9420 tskr1om 9468 tskr1om2 9469 uzrdg0i 12620 uzrdgsuci 12621 hashkf 12981 cshimadifsn 13426 cshimadifsn0 13427 shftfn 13661 phimullem 15322 imasaddvallem 16012 imasvscaval 16021 frmdss2 17223 lcomfsupp 18726 lidlval 19013 rspval 19014 psrbagev1 19331 psgnghm 19745 frlmsslsp 19954 iscldtop 20709 2ndcomap 21071 qtoptop 21313 basqtop 21324 qtoprest 21330 kqfvima 21343 isr0 21350 kqreglem1 21354 kqnrmlem1 21356 kqnrmlem2 21357 elrnust 21838 ustbas 21841 uniiccdif 23152 eupap1 26503 fcoinver 28798 fnct 28876 mdetpmtr1 29217 fimaproj 29228 sseqf 29781 sseqfv2 29783 elorrvc 29852 bnj142OLD 30048 bnj930 30094 bnj1371 30351 bnj1497 30382 frrlem2 31025 filnetlem4 31546 heibor1lem 32778 diaf11N 35356 dibf11N 35468 dibclN 35469 dihintcl 35651 ismrc 36282 dnnumch1 36632 aomclem4 36645 aomclem6 36647 ntrclsfv1 37373 ntrneifv1 37397 icccncfext 38773 stoweidlem29 38922 stoweidlem59 38952 ovolval4lem1 39539 fnresfnco 39855 funcoressn 39856 fnbrafvb 39883 tz6.12-afv 39902 afvco2 39905 resfnfinfin 40339 plusfreseq 41562 dfrngc2 41764 dfringc2 41810 rngcresringcat 41822 mndpsuppss 41946 mndpfsupp 41951 |
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