Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funpartlem Structured version   Visualization version   Unicode version

Theorem funpartlem 30780
 Description: Lemma for funpartfun 30781. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
funpartlem Image Singleton
Distinct variable groups:   ,   ,

Proof of Theorem funpartlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3040 . 2 Image Singleton
2 ssnid 3989 . . . . 5
3 eleq2 2538 . . . . 5
42, 3mpbiri 241 . . . 4
5 n0i 3727 . . . . 5
6 snprc 4027 . . . . . . . 8
76biimpi 199 . . . . . . 7
87imaeq2d 5174 . . . . . 6
9 ima0 5189 . . . . . 6
108, 9syl6eq 2521 . . . . 5
115, 10nsyl2 132 . . . 4
124, 11syl 17 . . 3
1312exlimiv 1784 . 2
14 eleq1 2537 . . 3 Image Singleton Image Singleton
15 sneq 3969 . . . . . 6
1615imaeq2d 5174 . . . . 5
1716eqeq1d 2473 . . . 4
1817exbidv 1776 . . 3
19 vex 3034 . . . . 5
2019eldm 5037 . . . 4 Image Singleton Image Singleton
21 brxp 4870 . . . . . . . . . 10
2219, 21mpbiran 932 . . . . . . . . 9
23 elsingles 30756 . . . . . . . . 9
2422, 23bitri 257 . . . . . . . 8
2524anbi2i 708 . . . . . . 7 Image Singleton Image Singleton
26 brin 4445 . . . . . . 7 Image Singleton Image Singleton
27 19.42v 1842 . . . . . . 7 Image Singleton Image Singleton
2825, 26, 273bitr4i 285 . . . . . 6 Image Singleton Image Singleton
2928exbii 1726 . . . . 5 Image Singleton Image Singleton
30 excom 1944 . . . . 5 Image Singleton Image Singleton
3129, 30bitri 257 . . . 4 Image Singleton Image Singleton
32 exancom 1730 . . . . . 6 Image Singleton Image Singleton
33 snex 4641 . . . . . . 7
34 breq2 4399 . . . . . . 7 Image Singleton Image Singleton
3533, 34ceqsexv 3070 . . . . . 6 Image Singleton Image Singleton
3619, 33brco 5010 . . . . . . 7 Image Singleton Singleton Image
37 vex 3034 . . . . . . . . . 10
3819, 37brsingle 30755 . . . . . . . . 9 Singleton
3938anbi1i 709 . . . . . . . 8 Singleton Image Image
4039exbii 1726 . . . . . . 7 Singleton Image Image
41 snex 4641 . . . . . . . . 9
42 breq1 4398 . . . . . . . . 9 Image Image
4341, 42ceqsexv 3070 . . . . . . . 8 Image Image
4441, 33brimage 30764 . . . . . . . 8 Image
45 eqcom 2478 . . . . . . . 8
4643, 44, 453bitri 279 . . . . . . 7 Image
4736, 40, 463bitri 279 . . . . . 6 Image Singleton
4832, 35, 473bitri 279 . . . . 5 Image Singleton
4948exbii 1726 . . . 4 Image Singleton
5020, 31, 493bitri 279 . . 3 Image Singleton
5114, 18, 50vtoclbg 3094 . 2 Image Singleton
521, 13, 51pm5.21nii 360 1 Image Singleton
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  cvv 3031   cin 3389  c0 3722  csn 3959   class class class wbr 4395   cxp 4837   cdm 4839  cima 4842   ccom 4843  Singletoncsingle 30675  csingles 30676  Imagecimage 30677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700  df-image 30701 This theorem is referenced by:  funpartfun  30781  funpartfv  30783
 Copyright terms: Public domain W3C validator