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Theorem elrnmpt1 5121
 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 2961 . . . 4
2 id 21 . . . . . . 7
3 csbeq1a 3261 . . . . . . 7
42, 3eleq12d 2506 . . . . . 6
5 csbeq1a 3261 . . . . . . 7
65biantrud 495 . . . . . 6
74, 6bitr2d 247 . . . . 5
87equcoms 1694 . . . 4
91, 8spcev 3045 . . 3
10 df-rex 2713 . . . . . 6
11 nfv 1630 . . . . . . 7
12 nfcsb1v 3285 . . . . . . . . 9
1312nfcri 2568 . . . . . . . 8
14 nfcsb1v 3285 . . . . . . . . 9
1514nfeq2 2585 . . . . . . . 8
1613, 15nfan 1847 . . . . . . 7
175eqeq2d 2449 . . . . . . . 8
184, 17anbi12d 693 . . . . . . 7
1911, 16, 18cbvex 1984 . . . . . 6
2010, 19bitri 242 . . . . 5
21 eqeq1 2444 . . . . . . 7
2221anbi2d 686 . . . . . 6
2322exbidv 1637 . . . . 5
2420, 23syl5bb 250 . . . 4
25 rnmpt.1 . . . . 5
2625rnmpt 5118 . . . 4
2724, 26elab2g 3086 . . 3
289, 27syl5ibr 214 . 2
2928impcom 421 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  wrex 2708  csb 3253   cmpt 4268   crn 4881 This theorem is referenced by:  fliftel1  6034  minveclem4  19335  minvecolem4  22384  rexunirn  23980  totbndbnd  26500  rrnequiv  26546 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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 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-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  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-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891
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