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Theorem zfrep6 6767
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 4578 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 4568. (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 2309 . . . . . . 7  |-  ( E! y ph  ->  E. y ph )
21ralimi 2850 . . . . . 6  |-  ( A. x  e.  z  E! y ph  ->  A. x  e.  z  E. y ph )
3 rabid2 3035 . . . . . 6  |-  ( z  =  { x  e.  z  |  E. y ph }  <->  A. x  e.  z  E. y ph )
42, 3sylibr 212 . . . . 5  |-  ( A. x  e.  z  E! y ph  ->  z  =  { x  e.  z  |  E. y ph }
)
5 19.42v 1776 . . . . . . 7  |-  ( E. y ( x  e.  z  /\  ph )  <->  ( x  e.  z  /\  E. y ph ) )
65abbii 2591 . . . . . 6  |-  { x  |  E. y ( x  e.  z  /\  ph ) }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
7 dmopab 5223 . . . . . 6  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  |  E. y ( x  e.  z  /\  ph ) }
8 df-rab 2816 . . . . . 6  |-  { x  e.  z  |  E. y ph }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
96, 7, 83eqtr4i 2496 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  e.  z  |  E. y ph }
104, 9syl6reqr 2517 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  z )
11 vex 3112 . . . 4  |-  z  e. 
_V
1210, 11syl6eqel 2553 . . 3  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
13 eumo 2314 . . . . . . 7  |-  ( E! y ph  ->  E* y ph )
1413imim2i 14 . . . . . 6  |-  ( ( x  e.  z  ->  E! y ph )  -> 
( x  e.  z  ->  E* y ph ) )
15 moanimv 2352 . . . . . 6  |-  ( E* y ( x  e.  z  /\  ph )  <->  ( x  e.  z  ->  E* y ph ) )
1614, 15sylibr 212 . . . . 5  |-  ( ( x  e.  z  ->  E! y ph )  ->  E* y ( x  e.  z  /\  ph )
)
1716alimi 1634 . . . 4  |-  ( A. x ( x  e.  z  ->  E! y ph )  ->  A. x E* y ( x  e.  z  /\  ph )
)
18 df-ral 2812 . . . 4  |-  ( A. x  e.  z  E! y ph  <->  A. x ( x  e.  z  ->  E! y ph ) )
19 funopab 5627 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<-> 
A. x E* y
( x  e.  z  /\  ph ) )
2017, 18, 193imtr4i 266 . . 3  |-  ( A. x  e.  z  E! y ph  ->  Fun  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
21 funrnex 6766 . . 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 60 . 2  |-  ( A. x  e.  z  E! y ph  ->  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
23 nfra1 2838 . . 3  |-  F/ x A. x  e.  z  E! y ph
2410eleq2d 2527 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  x  e.  z ) )
25 opabid 4763 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<->  ( x  e.  z  /\  ph ) )
26 vex 3112 . . . . . . . . . 10  |-  x  e. 
_V
27 vex 3112 . . . . . . . . . 10  |-  y  e. 
_V
2826, 27opelrn 5244 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
2925, 28sylbir 213 . . . . . . . 8  |-  ( ( x  e.  z  /\  ph )  ->  y  e.  ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } )
3029ex 434 . . . . . . 7  |-  ( x  e.  z  ->  ( ph  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } ) )
3130impac 621 . . . . . 6  |-  ( ( x  e.  z  /\  ph )  ->  ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
3231eximi 1657 . . . . 5  |-  ( E. y ( x  e.  z  /\  ph )  ->  E. y ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
337abeq2i 2584 . . . . 5  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  E. y
( x  e.  z  /\  ph ) )
34 df-rex 2813 . . . . 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 266 . . . 4  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
3624, 35syl6bir 229 . . 3  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  z  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
)
3723, 36ralrimi 2857 . 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 4523 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
3938nfrn 5255 . . . . 5  |-  F/_ x ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4039nfeq2 2636 . . . 4  |-  F/ x  w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }
41 nfcv 2619 . . . . 5  |-  F/_ y
w
42 nfopab2 4524 . . . . . 6  |-  F/_ y { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4342nfrn 5255 . . . . 5  |-  F/_ y ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4441, 43rexeqf 3051 . . . 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 2891 . . 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 3195 . 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 60 1  |-  ( A. x  e.  z  E! y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   E!weu 2283   E*wmo 2284   {cab 2442   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   <.cop 4038   {copab 4514   dom cdm 5008   ran crn 5009   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  bnj865  34082
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