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Theorem zfrep6 5961
 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 4323 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 4313. (Contributed by NM, 10-Oct-2003.)
Assertion
Ref Expression
zfrep6
Distinct variable groups:   ,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem zfrep6
StepHypRef Expression
1 euex 2304 . . . . . . 7
21ralimi 2774 . . . . . 6
3 rabid2 2878 . . . . . 6
42, 3sylibr 204 . . . . 5
5 19.42v 1928 . . . . . . 7
65abbii 2548 . . . . . 6
7 dmopab 5073 . . . . . 6
8 df-rab 2707 . . . . . 6
96, 7, 83eqtr4i 2466 . . . . 5
104, 9syl6reqr 2487 . . . 4
11 vex 2952 . . . 4
1210, 11syl6eqel 2524 . . 3
13 eumo 2321 . . . . . . 7
1413imim2i 14 . . . . . 6
15 moanimv 2339 . . . . . 6
1614, 15sylibr 204 . . . . 5
1716alimi 1568 . . . 4
18 df-ral 2703 . . . 4
19 funopab 5479 . . . 4
2017, 18, 193imtr4i 258 . . 3
21 funrnex 5960 . . 3
2212, 20, 21sylc 58 . 2
23 nfra1 2749 . . 3
2410eleq2d 2503 . . . 4
25 opabid 4454 . . . . . . . . 9
26 vex 2952 . . . . . . . . . 10
27 vex 2952 . . . . . . . . . 10
2826, 27opelrn 5094 . . . . . . . . 9
2925, 28sylbir 205 . . . . . . . 8
3029ex 424 . . . . . . 7
3130impac 605 . . . . . 6
3231eximi 1585 . . . . 5
337abeq2i 2543 . . . . 5
34 df-rex 2704 . . . . 5
3532, 33, 343imtr4i 258 . . . 4
3624, 35syl6bir 221 . . 3
3723, 36ralrimi 2780 . 2
38 nfopab1 4267 . . . . . 6
3938nfrn 5105 . . . . 5
4039nfeq2 2583 . . . 4
41 nfcv 2572 . . . . 5
42 nfopab2 4268 . . . . . 6
4342nfrn 5105 . . . . 5
4441, 43rexeqf 2894 . . . 4
4540, 44ralbid 2716 . . 3
4645spcegv 3030 . 2
4722, 37, 46sylc 58 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  weu 2281  wmo 2282  cab 2422  wral 2698  wrex 2699  crab 2702  cvv 2949  cop 3810  copab 4258   cdm 4871   crn 4872   wfun 5441 This theorem is referenced by:  bnj865  29232 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455
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