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Theorem zfrep6 6780
 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 4518 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 4508. (Contributed by NM, 10-Oct-2003.)
Assertion
Ref Expression
zfrep6
Distinct variable groups:   ,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem zfrep6
StepHypRef Expression
1 euex 2343 . . . . . . 7
21ralimi 2796 . . . . . 6
3 rabid2 2954 . . . . . 6
42, 3sylibr 217 . . . . 5
5 19.42v 1842 . . . . . . 7
65abbii 2587 . . . . . 6
7 dmopab 5051 . . . . . 6
8 df-rab 2765 . . . . . 6
96, 7, 83eqtr4i 2503 . . . . 5
104, 9syl6reqr 2524 . . . 4
11 vex 3034 . . . 4
1210, 11syl6eqel 2557 . . 3
13 eumo 2348 . . . . . . 7
1413imim2i 16 . . . . . 6
15 moanimv 2380 . . . . . 6
1614, 15sylibr 217 . . . . 5
1716alimi 1692 . . . 4
18 df-ral 2761 . . . 4
19 funopab 5622 . . . 4
2017, 18, 193imtr4i 274 . . 3
21 funrnex 6779 . . 3
2212, 20, 21sylc 61 . 2
23 nfra1 2785 . . 3
2410eleq2d 2534 . . . 4
25 opabid 4708 . . . . . . . . 9
26 vex 3034 . . . . . . . . . 10
27 vex 3034 . . . . . . . . . 10
2826, 27opelrn 5072 . . . . . . . . 9
2925, 28sylbir 218 . . . . . . . 8
3029ex 441 . . . . . . 7
3130impac 633 . . . . . 6
3231eximi 1715 . . . . 5
337abeq2i 2583 . . . . 5
34 df-rex 2762 . . . . 5
3532, 33, 343imtr4i 274 . . . 4
3624, 35syl6bir 237 . . 3
3723, 36ralrimi 2800 . 2
38 nfopab1 4462 . . . . . 6
3938nfrn 5083 . . . . 5
4039nfeq2 2627 . . . 4
41 nfcv 2612 . . . . 5
42 nfopab2 4463 . . . . . 6
4342nfrn 5083 . . . . 5
4441, 43rexeqf 2970 . . . 4
4540, 44ralbid 2826 . . 3
4645spcegv 3121 . 2
4722, 37, 46sylc 61 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376  wal 1450   wceq 1452  wex 1671   wcel 1904  weu 2319  wmo 2320  cab 2457  wral 2756  wrex 2757  crab 2760  cvv 3031  cop 3965  copab 4453   cdm 4839   crn 4840   wfun 5583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597 This theorem is referenced by:  bnj865  29806
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