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Theorem imgfldref2 25064
 Description: If is a reflexive relation and a part of its field, is a part of the image of by . (Contributed by FL, 3-Jul-2009.)
Assertion
Ref Expression
imgfldref2
Distinct variable groups:   ,   ,

Proof of Theorem imgfldref2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfra1 2593 . . . . 5
2 nfv 1605 . . . . 5
31, 2nfan 1771 . . . 4
4 ssel 3174 . . . . . . . . 9
5 rsp 2603 . . . . . . . . . 10
6 breq1 4026 . . . . . . . . . . . 12
76rspcev 2884 . . . . . . . . . . 11
87expcom 424 . . . . . . . . . 10
95, 8syl6com 31 . . . . . . . . 9
104, 9syl6com 31 . . . . . . . 8
1110com24 81 . . . . . . 7
1211pm2.43i 43 . . . . . 6
1312com3l 75 . . . . 5
1413imp 418 . . . 4
153, 14alrimi 1745 . . 3
16 ssab 3243 . . 3
1715, 16sylibr 203 . 2
18 dfima2 5014 . 2
1917, 18syl6sseqr 3225 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wal 1527   wcel 1684  cab 2269  wral 2543  wrex 2544   wss 3152  cuni 3827   class class class wbr 4023  cima 4692 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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