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Theorem opabiotafun 5934
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 5627 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  { y  |  ph }  =  {
y } }  <->  A. x E* y { y  | 
ph }  =  {
y } )
2 mo2icl 3278 . . . . 5  |-  ( A. z ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )  ->  E* z { y  |  ph }  =  { z } )
3 unieq 4259 . . . . . 6  |-  ( { y  |  ph }  =  { z }  ->  U. { y  |  ph }  =  U. { z } )
4 vex 3112 . . . . . . 7  |-  z  e. 
_V
54unisn 4266 . . . . . 6  |-  U. {
z }  =  z
63, 5syl6req 2515 . . . . 5  |-  ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )
72, 6mpg 1621 . . . 4  |-  E* z { y  |  ph }  =  { z }
8 nfv 1708 . . . . 5  |-  F/ z { y  |  ph }  =  { y }
9 nfab1 2621 . . . . . 6  |-  F/_ y { y  |  ph }
109nfeq1 2634 . . . . 5  |-  F/ y { y  |  ph }  =  { z }
11 sneq 4042 . . . . . 6  |-  ( y  =  z  ->  { y }  =  { z } )
1211eqeq2d 2471 . . . . 5  |-  ( y  =  z  ->  ( { y  |  ph }  =  { y } 
<->  { y  |  ph }  =  { z } ) )
138, 10, 12cbvmo 2323 . . . 4  |-  ( E* y { y  | 
ph }  =  {
y }  <->  E* z { y  |  ph }  =  { z } )
147, 13mpbir 209 . . 3  |-  E* y { y  |  ph }  =  { y }
151, 14mpgbir 1623 . 2  |-  Fun  { <. x ,  y >.  |  { y  |  ph }  =  { y } }
16 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
1716funeqi 5614 . 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 1395   E*wmo 2284   {cab 2442   {csn 4032   U.cuni 4251   {copab 4514   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-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
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-rab 2816  df-v 3111  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-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-fun 5596
This theorem is referenced by:  opabiota  5936
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