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Theorem elrnmptg 5252
 Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1
Assertion
Ref Expression
elrnmptg
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem elrnmptg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rnmpt.1 . . . 4
21rnmpt 5248 . . 3
32eleq2i 2545 . 2
4 r19.29 2997 . . . . 5
5 eleq1 2539 . . . . . . . 8
65biimparc 487 . . . . . . 7
7 elex 3122 . . . . . . 7
86, 7syl 16 . . . . . 6
98rexlimivw 2952 . . . . 5
104, 9syl 16 . . . 4
1110ex 434 . . 3
12 eqeq1 2471 . . . . 5
1312rexbidv 2973 . . . 4
1413elab3g 3256 . . 3
1511, 14syl 16 . 2
163, 15syl5bb 257 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  cab 2452  wral 2814  wrex 2815  cvv 3113   cmpt 4505   crn 5000 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010 This theorem is referenced by:  elrnmpti  5253
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