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Theorem zfrep6 6780
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 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  |-  ( 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 2343 . . . . . . 7  |-  ( E! y ph  ->  E. y ph )
21ralimi 2796 . . . . . 6  |-  ( A. x  e.  z  E! y ph  ->  A. x  e.  z  E. y ph )
3 rabid2 2954 . . . . . 6  |-  ( z  =  { x  e.  z  |  E. y ph }  <->  A. x  e.  z  E. y ph )
42, 3sylibr 217 . . . . 5  |-  ( A. x  e.  z  E! y ph  ->  z  =  { x  e.  z  |  E. y ph }
)
5 19.42v 1842 . . . . . . 7  |-  ( E. y ( x  e.  z  /\  ph )  <->  ( x  e.  z  /\  E. y ph ) )
65abbii 2587 . . . . . 6  |-  { x  |  E. y ( x  e.  z  /\  ph ) }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
7 dmopab 5051 . . . . . 6  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  |  E. y ( x  e.  z  /\  ph ) }
8 df-rab 2765 . . . . . 6  |-  { x  e.  z  |  E. y ph }  =  {
x  |  ( x  e.  z  /\  E. y ph ) }
96, 7, 83eqtr4i 2503 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  { x  e.  z  |  E. y ph }
104, 9syl6reqr 2524 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  =  z )
11 vex 3034 . . . 4  |-  z  e. 
_V
1210, 11syl6eqel 2557 . . 3  |-  ( A. x  e.  z  E! y ph  ->  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
13 eumo 2348 . . . . . . 7  |-  ( E! y ph  ->  E* y ph )
1413imim2i 16 . . . . . 6  |-  ( ( x  e.  z  ->  E! y ph )  -> 
( x  e.  z  ->  E* y ph ) )
15 moanimv 2380 . . . . . 6  |-  ( E* y ( x  e.  z  /\  ph )  <->  ( x  e.  z  ->  E* y ph ) )
1614, 15sylibr 217 . . . . 5  |-  ( ( x  e.  z  ->  E! y ph )  ->  E* y ( x  e.  z  /\  ph )
)
1716alimi 1692 . . . 4  |-  ( A. x ( x  e.  z  ->  E! y ph )  ->  A. x E* y ( x  e.  z  /\  ph )
)
18 df-ral 2761 . . . 4  |-  ( A. x  e.  z  E! y ph  <->  A. x ( x  e.  z  ->  E! y ph ) )
19 funopab 5622 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<-> 
A. x E* y
( x  e.  z  /\  ph ) )
2017, 18, 193imtr4i 274 . . 3  |-  ( A. x  e.  z  E! y ph  ->  Fun  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
21 funrnex 6779 . . 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 61 . 2  |-  ( A. x  e.  z  E! y ph  ->  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  e.  _V )
23 nfra1 2785 . . 3  |-  F/ x A. x  e.  z  E! y ph
2410eleq2d 2534 . . . 4  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  x  e.  z ) )
25 opabid 4708 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } 
<->  ( x  e.  z  /\  ph ) )
26 vex 3034 . . . . . . . . . 10  |-  x  e. 
_V
27 vex 3034 . . . . . . . . . 10  |-  y  e. 
_V
2826, 27opelrn 5072 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } )
2925, 28sylbir 218 . . . . . . . 8  |-  ( ( x  e.  z  /\  ph )  ->  y  e.  ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) } )
3029ex 441 . . . . . . 7  |-  ( x  e.  z  ->  ( ph  ->  y  e.  ran  {
<. x ,  y >.  |  ( x  e.  z  /\  ph ) } ) )
3130impac 633 . . . . . 6  |-  ( ( x  e.  z  /\  ph )  ->  ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
3231eximi 1715 . . . . 5  |-  ( E. y ( x  e.  z  /\  ph )  ->  E. y ( y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  /\  ph ) )
337abeq2i 2583 . . . . 5  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  <->  E. y
( x  e.  z  /\  ph ) )
34 df-rex 2762 . . . . 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 274 . . . 4  |-  ( x  e.  dom  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
3624, 35syl6bir 237 . . 3  |-  ( A. x  e.  z  E! y ph  ->  ( x  e.  z  ->  E. y  e.  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) } ph )
)
3723, 36ralrimi 2800 . 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 4462 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
3938nfrn 5083 . . . . 5  |-  F/_ x ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4039nfeq2 2627 . . . 4  |-  F/ x  w  =  ran  { <. x ,  y >.  |  ( x  e.  z  /\  ph ) }
41 nfcv 2612 . . . . 5  |-  F/_ y
w
42 nfopab2 4463 . . . . . 6  |-  F/_ y { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4342nfrn 5083 . . . . 5  |-  F/_ y ran  { <. x ,  y
>.  |  ( x  e.  z  /\  ph ) }
4441, 43rexeqf 2970 . . . 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 2826 . . 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 3121 . 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 61 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 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   E*wmo 2320   {cab 2457   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   <.cop 3965   {copab 4453   dom cdm 4839   ran crn 4840   Fun 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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