MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabiotafun Structured version   Unicode version

Theorem opabiotafun 6528
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 5478 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  { y  |  ph }  =  {
y } }  <->  A. x E* y { y  | 
ph }  =  {
y } )
2 mo2icl 3105 . . . . 5  |-  ( A. z ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )  ->  E* z { y  |  ph }  =  { z } )
3 unieq 4016 . . . . . 6  |-  ( { y  |  ph }  =  { z }  ->  U. { y  |  ph }  =  U. { z } )
4 vex 2951 . . . . . . 7  |-  z  e. 
_V
54unisn 4023 . . . . . 6  |-  U. {
z }  =  z
63, 5syl6req 2484 . . . . 5  |-  ( { y  |  ph }  =  { z }  ->  z  =  U. { y  |  ph } )
72, 6mpg 1557 . . . 4  |-  E* z { y  |  ph }  =  { z }
8 nfv 1629 . . . . 5  |-  F/ z { y  |  ph }  =  { y }
9 nfab1 2573 . . . . . 6  |-  F/_ y { y  |  ph }
109nfeq1 2580 . . . . 5  |-  F/ y { y  |  ph }  =  { z }
11 sneq 3817 . . . . . 6  |-  ( y  =  z  ->  { y }  =  { z } )
1211eqeq2d 2446 . . . . 5  |-  ( y  =  z  ->  ( { y  |  ph }  =  { y } 
<->  { y  |  ph }  =  { z } ) )
138, 10, 12cbvmo 2317 . . . 4  |-  ( E* y { y  | 
ph }  =  {
y }  <->  E* z { y  |  ph }  =  { z } )
147, 13mpbir 201 . . 3  |-  E* y { y  |  ph }  =  { y }
151, 14mpgbir 1559 . 2  |-  Fun  { <. x ,  y >.  |  { y  |  ph }  =  { y } }
16 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
1716funeqi 5466 . 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 1652   E*wmo 2281   {cab 2421   {csn 3806   U.cuni 4007   {copab 4257   Fun wfun 5440
This theorem is referenced by:  opabiota  6530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-fun 5448
  Copyright terms: Public domain W3C validator