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Theorem funpartfun 25781
 Description: The functional part of is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartfun Funpart

Proof of Theorem funpartfun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5167 . 2 Image Singleton
2 vex 2952 . . . . . . 7
32brres 5145 . . . . . 6 Image Singleton Image Singleton
43simplbi 447 . . . . 5 Image Singleton
5 vex 2952 . . . . . . . 8
65brres 5145 . . . . . . 7 Image Singleton Image Singleton
7 ancom 438 . . . . . . . 8 Image Singleton Image Singleton
8 funpartlem 25780 . . . . . . . . 9 Image Singleton
98anbi1i 677 . . . . . . . 8 Image Singleton
107, 9bitri 241 . . . . . . 7 Image Singleton
116, 10bitri 241 . . . . . 6 Image Singleton
12 df-br 4206 . . . . . . . . . . 11
13 df-br 4206 . . . . . . . . . . 11
1412, 13anbi12i 679 . . . . . . . . . 10
15 vex 2952 . . . . . . . . . . . 12
1615, 5elimasn 5222 . . . . . . . . . . 11
1715, 2elimasn 5222 . . . . . . . . . . 11
1816, 17anbi12i 679 . . . . . . . . . 10
1914, 18bitr4i 244 . . . . . . . . 9
20 eleq2 2497 . . . . . . . . . . 11
21 eleq2 2497 . . . . . . . . . . 11
2220, 21anbi12d 692 . . . . . . . . . 10
23 elsn 3822 . . . . . . . . . . 11
24 elsn 3822 . . . . . . . . . . 11
25 equtr2 1700 . . . . . . . . . . 11
2623, 24, 25syl2anb 466 . . . . . . . . . 10
2722, 26syl6bi 220 . . . . . . . . 9
2819, 27syl5bi 209 . . . . . . . 8
2928exlimiv 1644 . . . . . . 7
3029impl 604 . . . . . 6
3111, 30sylanb 459 . . . . 5 Image Singleton
324, 31sylan2 461 . . . 4 Image Singleton Image Singleton
3332gen2 1556 . . 3 Image Singleton Image Singleton
3433ax-gen 1555 . 2 Image Singleton Image Singleton
35 df-funpart 25711 . . . 4 Funpart Image Singleton
3635funeqi 5467 . . 3 Funpart Image Singleton
37 dffun2 5457 . . 3 Image Singleton Image Singleton Image Singleton Image Singleton
3836, 37bitri 241 . 2 Funpart Image Singleton Image Singleton Image Singleton
391, 34, 38mpbir2an 887 1 Funpart
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  cvv 2949   cin 3312  csn 3807  cop 3810   class class class wbr 4205   cxp 4869   cdm 4871   cres 4873  cima 4874   ccom 4875   wrel 4876   wfun 5441  Singletoncsingle 25675  csingles 25676  Imagecimage 25677  Funpartcfunpart 25686 This theorem is referenced by:  fullfunfnv  25784  fullfunfv  25785 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-eprel 4487  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-1st 6342  df-2nd 6343  df-symdif 25656  df-txp 25691  df-singleton 25699  df-singles 25700  df-image 25701  df-funpart 25711
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