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Theorem zfrep6 5927
Description: A version of the Axiom of Replacement. Normally  ph would have free variables  x and  y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4290 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 4280. (Contributed by NM, 10-Oct-2003.)
Assertion
Ref Expression
zfrep6  |-  ( A. x  e.  z  E! y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
Distinct variable groups:    ph, w    x, y, z, w
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem zfrep6
StepHypRef Expression
1 euex 2277 . . . . . . 7  |-  ( E! y ph  ->  E. y ph )
21ralimi 2741 . . . . . 6  |-  ( A. x  e.  z  E! y ph  ->  A. x  e.  z  E. y ph )
3 rabid2 2845 . . . . . 6  |-  ( z  =  { x  e.  z  |  E. y ph }  <->  A. x  e.  z  E. y ph )
42, 3sylibr 204 . . . . 5  |-  ( A. x  e.  z  E! y ph  ->  z  =  { x  e.  z  |  E. y ph }
)
5 19.42v 1924 . . . . . . 7  |-  ( E. y ( x  e.  z  /\  ph )  <->  ( x  e.  z  /\  E. y ph ) )
65abbii 2516 . . . . . 6  |-  { x  |  E. y ( x  e.  z  /\  ph ) }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
7 dmopab 5039 . . . . . 6  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  |  E. y ( x  e.  z  /\  ph ) }
8 df-rab 2675 . . . . . 6  |-  { x  e.  z  |  E. y ph }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
96, 7, 83eqtr4i 2434 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  e.  z  |  E. y ph }
104, 9syl6reqr 2455 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  z )
11 vex 2919 . . . 4  |-  z  e. 
_V
1210, 11syl6eqel 2492 . . 3  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
13 eumo 2294 . . . . . . 7  |-  ( E! y ph  ->  E* y ph )
1413imim2i 14 . . . . . 6  |-  ( ( x  e.  z  ->  E! y ph )  -> 
( x  e.  z  ->  E* y ph ) )
15 moanimv 2312 . . . . . 6  |-  ( E* y ( x  e.  z  /\  ph )  <->  ( x  e.  z  ->  E* y ph ) )
1614, 15sylibr 204 . . . . 5  |-  ( ( x  e.  z  ->  E! y ph )  ->  E* y ( x  e.  z  /\  ph )
)
1716alimi 1565 . . . 4  |-  ( A. x ( x  e.  z  ->  E! y ph )  ->  A. x E* y ( x  e.  z  /\  ph )
)
18 df-ral 2671 . . . 4  |-  ( A. x  e.  z  E! y ph  <->  A. x ( x  e.  z  ->  E! y ph ) )
19 funopab 5445 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<-> 
A. x E* y
( x  e.  z  /\  ph ) )
2017, 18, 193imtr4i 258 . . 3  |-  ( A. x  e.  z  E! y ph  ->  Fun  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
21 funrnex 5926 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  e.  _V  ->  ( Fun  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
)
2212, 20, 21sylc 58 . 2  |-  ( A. x  e.  z  E! y ph  ->  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
23 nfra1 2716 . . 3  |-  F/ x A. x  e.  z  E! y ph
2410eleq2d 2471 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  x  e.  z ) )
25 opabid 4421 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<->  ( x  e.  z  /\  ph ) )
26 vex 2919 . . . . . . . . . 10  |-  x  e. 
_V
27 vex 2919 . . . . . . . . . 10  |-  y  e. 
_V
2826, 27opelrn 5060 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
2925, 28sylbir 205 . . . . . . . 8  |-  ( ( x  e.  z  /\  ph )  ->  y  e.  ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } )
3029ex 424 . . . . . . 7  |-  ( x  e.  z  ->  ( ph  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } ) )
3130impac 605 . . . . . 6  |-  ( ( x  e.  z  /\  ph )  ->  ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
3231eximi 1582 . . . . 5  |-  ( E. y ( x  e.  z  /\  ph )  ->  E. y ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
337abeq2i 2511 . . . . 5  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  E. y
( x  e.  z  /\  ph ) )
34 df-rex 2672 . . . . 5  |-  ( E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph  <->  E. y ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
3532, 33, 343imtr4i 258 . . . 4  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
3624, 35syl6bir 221 . . 3  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  z  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
)
3723, 36ralrimi 2747 . 2  |-  ( A. x  e.  z  E! y ph  ->  A. x  e.  z  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
38 nfopab1 4234 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
3938nfrn 5071 . . . . 5  |-  F/_ x ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4039nfeq2 2551 . . . 4  |-  F/ x  w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }
41 nfcv 2540 . . . . 5  |-  F/_ y
w
42 nfopab2 4235 . . . . . 6  |-  F/_ y { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4342nfrn 5071 . . . . 5  |-  F/_ y ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4441, 43rexeqf 2861 . . . 4  |-  ( w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  ( E. y  e.  w  ph  <->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph ) )
4540, 44ralbid 2684 . . 3  |-  ( w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  ( A. x  e.  z  E. y  e.  w  ph  <->  A. x  e.  z  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
)
4645spcegv 2997 . 2  |-  ( ran 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  e.  _V  ->  ( A. x  e.  z  E. y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph  ->  E. w A. x  e.  z  E. y  e.  w  ph ) )
4722, 37, 46sylc 58 1  |-  ( A. x  e.  z  E! y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254   E*wmo 2255   {cab 2390   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916   <.cop 3777   {copab 4225   dom cdm 4837   ran crn 4838   Fun wfun 5407
This theorem is referenced by:  bnj865  29000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
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