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Theorem opabiotafun 5752
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 5451 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  { y  |  ph }  =  {
y } }  <->  A. x E* y { y  | 
ph }  =  {
y } )
2 mo2icl 3138 . . . . 5  |-  ( A. z ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )  ->  E* z { y  |  ph }  =  { z } )
3 unieq 4099 . . . . . 6  |-  ( { y  |  ph }  =  { z }  ->  U. { y  |  ph }  =  U. { z } )
4 vex 2975 . . . . . . 7  |-  z  e. 
_V
54unisn 4106 . . . . . 6  |-  U. {
z }  =  z
63, 5syl6req 2492 . . . . 5  |-  ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )
72, 6mpg 1593 . . . 4  |-  E* z { y  |  ph }  =  { z }
8 nfv 1673 . . . . 5  |-  F/ z { y  |  ph }  =  { y }
9 nfab1 2581 . . . . . 6  |-  F/_ y { y  |  ph }
109nfeq1 2588 . . . . 5  |-  F/ y { y  |  ph }  =  { z }
11 sneq 3887 . . . . . 6  |-  ( y  =  z  ->  { y }  =  { z } )
1211eqeq2d 2454 . . . . 5  |-  ( y  =  z  ->  ( { y  |  ph }  =  { y } 
<->  { y  |  ph }  =  { z } ) )
138, 10, 12cbvmo 2297 . . . 4  |-  ( E* y { y  | 
ph }  =  {
y }  <->  E* z { y  |  ph }  =  { z } )
147, 13mpbir 209 . . 3  |-  E* y { y  |  ph }  =  { y }
151, 14mpgbir 1595 . 2  |-  Fun  { <. x ,  y >.  |  { y  |  ph }  =  { y } }
16 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
1716funeqi 5438 . 2  |-  ( Fun 
F  <->  Fun  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
)
1815, 17mpbir 209 1  |-  Fun  F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   E*wmo 2254   {cab 2429   {csn 3877   U.cuni 4091   {copab 4349   Fun wfun 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-fun 5420
This theorem is referenced by:  opabiota  5754
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