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Theorem elrnmpt1 5256
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1
Assertion
Ref Expression
elrnmpt1

Proof of Theorem elrnmpt1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . 4
2 id 22 . . . . . . 7
3 csbeq1a 3443 . . . . . . 7
42, 3eleq12d 2539 . . . . . 6
5 csbeq1a 3443 . . . . . . 7
65biantrud 507 . . . . . 6
74, 6bitr2d 254 . . . . 5
87equcoms 1795 . . . 4
91, 8spcev 3201 . . 3
10 df-rex 2813 . . . . . 6
11 nfv 1707 . . . . . . 7
12 nfcsb1v 3450 . . . . . . . . 9
1312nfcri 2612 . . . . . . . 8
14 nfcsb1v 3450 . . . . . . . . 9
1514nfeq2 2636 . . . . . . . 8
1613, 15nfan 1928 . . . . . . 7
175eqeq2d 2471 . . . . . . . 8
184, 17anbi12d 710 . . . . . . 7
1911, 16, 18cbvex 2022 . . . . . 6
2010, 19bitri 249 . . . . 5
21 eqeq1 2461 . . . . . . 7
2221anbi2d 703 . . . . . 6
2322exbidv 1714 . . . . 5
2420, 23syl5bb 257 . . . 4
25 rnmpt.1 . . . . 5
2625rnmpt 5253 . . . 4
2724, 26elab2g 3248 . . 3
289, 27syl5ibr 221 . 2
2928impcom 430 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  [_csb 3434  e.cmpt 4510  rancrn 5005
This theorem is referenced by:  fliftel1  6208  minveclem4  21847  minvecolem4  25796  rexunirn  27390  totbndbnd  30285  rrnequiv  30331  suprnmpt  31451  fourierdlem31  31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-mpt 4512  df-cnv 5012  df-dm 5014  df-rn 5015
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