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Theorem rexrn 5550
 Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1
Assertion
Ref Expression
rexrn
Distinct variable groups:   ,,   ,,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem rexrn
StepHypRef Expression
1 funfvex 5431 . . 3
21funfni 5218 . 2
3 fvelrnb 5462 . . 3
4 eqcom 2139 . . . 4
54rexbii 2440 . . 3
63, 5syl6bb 195 . 2
7 rexrn.1 . . 3
87adantl 275 . 2
92, 6, 8rexxfr2d 4381 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1331   wcel 1480  wrex 2415  cvv 2681   crn 4535   wfn 5113  cfv 5118 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126 This theorem is referenced by:  elrnrexdm  5552  rexrnmpt  5556  cbvexfo  5680  rexanuz  10753  lmbr2  12372  lmff  12407
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