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Theorem opabiotafun 6474
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 5428 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  { y  |  ph }  =  {
y } }  <->  A. x E* y { y  | 
ph }  =  {
y } )
2 mo2icl 3058 . . . . 5  |-  ( A. z ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )  ->  E* z { y  |  ph }  =  { z } )
3 unieq 3968 . . . . . 6  |-  ( { y  |  ph }  =  { z }  ->  U. { y  |  ph }  =  U. { z } )
4 vex 2904 . . . . . . 7  |-  z  e. 
_V
54unisn 3975 . . . . . 6  |-  U. {
z }  =  z
63, 5syl6req 2438 . . . . 5  |-  ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )
72, 6mpg 1554 . . . 4  |-  E* z { y  |  ph }  =  { z }
8 nfv 1626 . . . . 5  |-  F/ z { y  |  ph }  =  { y }
9 nfab1 2527 . . . . . 6  |-  F/_ y { y  |  ph }
109nfeq1 2534 . . . . 5  |-  F/ y { y  |  ph }  =  { z }
11 sneq 3770 . . . . . 6  |-  ( y  =  z  ->  { y }  =  { z } )
1211eqeq2d 2400 . . . . 5  |-  ( y  =  z  ->  ( { y  |  ph }  =  { y } 
<->  { y  |  ph }  =  { z } ) )
138, 10, 12cbvmo 2277 . . . 4  |-  ( E* y { y  | 
ph }  =  {
y }  <->  E* z { y  |  ph }  =  { z } )
147, 13mpbir 201 . . 3  |-  E* y { y  |  ph }  =  { y }
151, 14mpgbir 1556 . 2  |-  Fun  { <. x ,  y >.  |  { y  |  ph }  =  { y } }
16 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
1716funeqi 5416 . 2  |-  ( Fun 
F  <->  Fun  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
)
1815, 17mpbir 201 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   E*wmo 2241   {cab 2375   {csn 3759   U.cuni 3959   {copab 4208   Fun wfun 5390
This theorem is referenced by:  opabiota  6476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-fun 5398
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