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Theorem opabiotafun 6291
Description: Define a function whose value is "the unique  y such that  ph ( x ,  y )". (Contributed by NM, 19-May-2015.)
Hypothesis
Ref Expression
opabiota.1  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
Assertion
Ref Expression
opabiotafun  |-  Fun  F
Distinct variable group:    x, y, F
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabiotafun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 funopab 5287 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  { y  |  ph }  =  {
y } }  <->  A. x E* y { y  | 
ph }  =  {
y } )
2 mo2icl 2944 . . . . 5  |-  ( A. z ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )  ->  E* z { y  |  ph }  =  { z } )
3 unieq 3836 . . . . . 6  |-  ( { y  |  ph }  =  { z }  ->  U. { y  |  ph }  =  U. { z } )
4 vex 2791 . . . . . . 7  |-  z  e. 
_V
54unisn 3843 . . . . . 6  |-  U. {
z }  =  z
63, 5syl6req 2332 . . . . 5  |-  ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )
72, 6mpg 1535 . . . 4  |-  E* z { y  |  ph }  =  { z }
8 nfv 1605 . . . . 5  |-  F/ z { y  |  ph }  =  { y }
9 nfab1 2421 . . . . . 6  |-  F/_ y { y  |  ph }
109nfeq1 2428 . . . . 5  |-  F/ y { y  |  ph }  =  { z }
11 sneq 3651 . . . . . 6  |-  ( y  =  z  ->  { y }  =  { z } )
1211eqeq2d 2294 . . . . 5  |-  ( y  =  z  ->  ( { y  |  ph }  =  { y } 
<->  { y  |  ph }  =  { z } ) )
138, 10, 12cbvmo 2180 . . . 4  |-  ( E* y { y  | 
ph }  =  {
y }  <->  E* z { y  |  ph }  =  { z } )
147, 13mpbir 200 . . 3  |-  E* y { y  |  ph }  =  { y }
151, 14mpgbir 1537 . 2  |-  Fun  { <. x ,  y >.  |  { y  |  ph }  =  { y } }
16 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
1716funeqi 5275 . 2  |-  ( Fun 
F  <->  Fun  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
)
1815, 17mpbir 200 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   E*wmo 2144   {cab 2269   {csn 3640   U.cuni 3827   {copab 4076   Fun wfun 5249
This theorem is referenced by:  opabiota  6293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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