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Theorem zfrep6 6768
Description: A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4573 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4563. (Contributed by NM, 10-Oct-2003.)
Assertion
Ref Expression
zfrep6
Distinct variable groups:   ,   , , ,

Proof of Theorem zfrep6
StepHypRef Expression
1 euex 2308 . . . . . . 7
21ralimi 2850 . . . . . 6
3 rabid2 3035 . . . . . 6
42, 3sylibr 212 . . . . 5
5 19.42v 1775 . . . . . . 7
65abbii 2591 . . . . . 6
7 dmopab 5218 . . . . . 6
8 df-rab 2816 . . . . . 6
96, 7, 83eqtr4i 2496 . . . . 5
104, 9syl6reqr 2517 . . . 4
11 vex 3112 . . . 4
1210, 11syl6eqel 2553 . . 3
13 eumo 2313 . . . . . . 7
1413imim2i 14 . . . . . 6
15 moanimv 2352 . . . . . 6
1614, 15sylibr 212 . . . . 5
1716alimi 1633 . . . 4
18 df-ral 2812 . . . 4
19 funopab 5626 . . . 4
2017, 18, 193imtr4i 266 . . 3
21 funrnex 6767 . . 3
2212, 20, 21sylc 60 . 2
23 nfra1 2838 . . 3
2410eleq2d 2527 . . . 4
25 opabid 4759 . . . . . . . . 9
26 vex 3112 . . . . . . . . . 10
27 vex 3112 . . . . . . . . . 10
2826, 27opelrn 5239 . . . . . . . . 9
2925, 28sylbir 213 . . . . . . . 8
3029ex 434 . . . . . . 7
3130impac 621 . . . . . 6
3231eximi 1656 . . . . 5
337abeq2i 2584 . . . . 5
34 df-rex 2813 . . . . 5
3532, 33, 343imtr4i 266 . . . 4
3624, 35syl6bir 229 . . 3
3723, 36ralrimi 2857 . 2
38 nfopab1 4518 . . . . . 6
3938nfrn 5250 . . . . 5
4039nfeq2 2636 . . . 4
41 nfcv 2619 . . . . 5
42 nfopab2 4519 . . . . . 6
4342nfrn 5250 . . . . 5
4441, 43rexeqf 3051 . . . 4
4540, 44ralbid 2891 . . 3
4645spcegv 3195 . 2
4722, 37, 46sylc 60 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  E*wmo 2283  {cab 2442  A.wral 2807  E.wrex 2808  {crab 2811   cvv 3109  <.cop 4035  {copab 4509  domcdm 5004  rancrn 5005  Funwfun 5587
This theorem is referenced by:  bnj865  33981
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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