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Theorem funpartfv 25792
 Description: The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartfv Funpart

Proof of Theorem funpartfv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-funpart 25720 . . 3 Funpart Image Singleton
21fveq1i 5731 . 2 Funpart Image Singleton
3 fvres 5747 . . 3 Image Singleton Image Singleton
4 nfvres 5762 . . . 4 Image Singleton Image Singleton
5 funpartlem 25789 . . . . . . . . 9 Image Singleton
6 eusn 3882 . . . . . . . . 9
75, 6bitr4i 245 . . . . . . . 8 Image Singleton
8 vex 2961 . . . . . . . . . . 11
9 elimasng 5232 . . . . . . . . . . 11
108, 9mpan2 654 . . . . . . . . . 10
11 df-br 4215 . . . . . . . . . 10
1210, 11syl6bbr 256 . . . . . . . . 9
1312eubidv 2291 . . . . . . . 8
147, 13syl5bb 250 . . . . . . 7 Image Singleton
1514notbid 287 . . . . . 6 Image Singleton
16 tz6.12-2 5721 . . . . . 6
1715, 16syl6bi 221 . . . . 5 Image Singleton
18 fvprc 5724 . . . . . 6
1918a1d 24 . . . . 5 Image Singleton
2017, 19pm2.61i 159 . . . 4 Image Singleton
214, 20eqtr4d 2473 . . 3 Image Singleton Image Singleton
223, 21pm2.61i 159 . 2 Image Singleton
232, 22eqtri 2458 1 Funpart
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178  wex 1551   wceq 1653   wcel 1726  weu 2283  cvv 2958   cin 3321  c0 3630  csn 3816  cop 3819   class class class wbr 4214   cxp 4878   cdm 4880   cres 4882  cima 4883   ccom 4884  cfv 5456  Singletoncsingle 25684  csingles 25685  Imagecimage 25686  Funpartcfunpart 25695 This theorem is referenced by:  fullfunfv  25794 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-singleton 25708  df-singles 25709  df-image 25710  df-funpart 25720
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