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Theorem zfrep6 5764
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 4157 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 4147. (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 2179 . . . . . . 7  |-  ( E! y ph  ->  E. y ph )
21ralimi 2631 . . . . . 6  |-  ( A. x  e.  z  E! y ph  ->  A. x  e.  z  E. y ph )
3 rabid2 2730 . . . . . 6  |-  ( z  =  { x  e.  z  |  E. y ph }  <->  A. x  e.  z  E. y ph )
42, 3sylibr 203 . . . . 5  |-  ( A. x  e.  z  E! y ph  ->  z  =  { x  e.  z  |  E. y ph }
)
5 19.42v 1858 . . . . . . 7  |-  ( E. y ( x  e.  z  /\  ph )  <->  ( x  e.  z  /\  E. y ph ) )
65abbii 2408 . . . . . 6  |-  { x  |  E. y ( x  e.  z  /\  ph ) }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
7 dmopab 4905 . . . . . 6  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  |  E. y ( x  e.  z  /\  ph ) }
8 df-rab 2565 . . . . . 6  |-  { x  e.  z  |  E. y ph }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
96, 7, 83eqtr4i 2326 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  e.  z  |  E. y ph }
104, 9syl6reqr 2347 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  z )
11 vex 2804 . . . 4  |-  z  e. 
_V
1210, 11syl6eqel 2384 . . 3  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
13 eumo 2196 . . . . . . 7  |-  ( E! y ph  ->  E* y ph )
1413imim2i 13 . . . . . 6  |-  ( ( x  e.  z  ->  E! y ph )  -> 
( x  e.  z  ->  E* y ph ) )
15 moanimv 2214 . . . . . 6  |-  ( E* y ( x  e.  z  /\  ph )  <->  ( x  e.  z  ->  E* y ph ) )
1614, 15sylibr 203 . . . . 5  |-  ( ( x  e.  z  ->  E! y ph )  ->  E* y ( x  e.  z  /\  ph )
)
1716alimi 1549 . . . 4  |-  ( A. x ( x  e.  z  ->  E! y ph )  ->  A. x E* y ( x  e.  z  /\  ph )
)
18 df-ral 2561 . . . 4  |-  ( A. x  e.  z  E! y ph  <->  A. x ( x  e.  z  ->  E! y ph ) )
19 funopab 5303 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<-> 
A. x E* y
( x  e.  z  /\  ph ) )
2017, 18, 193imtr4i 257 . . 3  |-  ( A. x  e.  z  E! y ph  ->  Fun  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
21 funrnex 5763 . . 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 56 . 2  |-  ( A. x  e.  z  E! y ph  ->  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
23 nfra1 2606 . . 3  |-  F/ x A. x  e.  z  E! y ph
2410eleq2d 2363 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  x  e.  z ) )
25 opabid 4287 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<->  ( x  e.  z  /\  ph ) )
26 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
27 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
2826, 27opelrn 4926 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
2925, 28sylbir 204 . . . . . . . 8  |-  ( ( x  e.  z  /\  ph )  ->  y  e.  ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } )
3029ex 423 . . . . . . 7  |-  ( x  e.  z  ->  ( ph  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } ) )
3130impac 604 . . . . . 6  |-  ( ( x  e.  z  /\  ph )  ->  ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
3231eximi 1566 . . . . 5  |-  ( E. y ( x  e.  z  /\  ph )  ->  E. y ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
337abeq2i 2403 . . . . 5  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  E. y
( x  e.  z  /\  ph ) )
34 df-rex 2562 . . . . 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 257 . . . 4  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
3624, 35syl6bir 220 . . 3  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  z  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
)
3723, 36ralrimi 2637 . 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 4101 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
3938nfrn 4937 . . . . 5  |-  F/_ x ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4039nfeq2 2443 . . . 4  |-  F/ x  w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }
41 nfcv 2432 . . . . 5  |-  F/_ y
w
42 nfopab2 4102 . . . . . 6  |-  F/_ y { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4342nfrn 4937 . . . . 5  |-  F/_ y ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4441, 43rexeqf 2746 . . . 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 2574 . . 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 2882 . 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 56 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 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   <.cop 3656   {copab 4092   dom cdm 4705   ran crn 4706   Fun wfun 5265
This theorem is referenced by:  bnj865  29271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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