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Theorem abrexex 6774
 Description: Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in the class expression substituted for , which can be thought of as (x). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 6142, funex 6140, fnex 6139, resfunexg 6137, and funimaexg 5670. See also abrexex2 6781. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
abrexex.1
Assertion
Ref Expression
abrexex
Distinct variable groups:   ,,   ,

Proof of Theorem abrexex
StepHypRef Expression
1 eqid 2457 . . 3
21rnmpt 5253 . 2
3 abrexex.1 . . . 4
43mptex 6143 . . 3
54rnex 6734 . 2
62, 5eqeltrri 2542 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  e.cmpt 4510  rancrn 5005 This theorem is referenced by:  ab2rexex  6791  kmlem10  8560  shftfval  12903  dvdsrval  17294  cmpsublem  19899  cmpsub  19900  ptrescn  20140  heibor1lem  30305  eldiophb  30690  pointsetN  35465 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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 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-ral 2812  df-rex 2813  df-reu 2814  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-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601
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