MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvopab5 Structured version   Unicode version

Theorem fvopab5 6526
Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab5.1  |-  F  =  { <. x ,  y
>.  |  ph }
fvopab5.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
fvopab5  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y ps ) )
Distinct variable groups:    x, y, A    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)    F( x, y)    V( x, y)

Proof of Theorem fvopab5
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 df-fv 5454 . . . 4  |-  ( F `
 A )  =  ( iota z A F z )
3 breq2 4208 . . . . 5  |-  ( z  =  y  ->  ( A F z  <->  A F
y ) )
4 nfcv 2571 . . . . . 6  |-  F/_ y A
5 fvopab5.1 . . . . . . 7  |-  F  =  { <. x ,  y
>.  |  ph }
6 nfopab2 4267 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
75, 6nfcxfr 2568 . . . . . 6  |-  F/_ y F
8 nfcv 2571 . . . . . 6  |-  F/_ y
z
94, 7, 8nfbr 4248 . . . . 5  |-  F/ y  A F z
10 nfv 1629 . . . . 5  |-  F/ z  A F y
113, 9, 10cbviota 5415 . . . 4  |-  ( iota z A F z )  =  ( iota y A F y )
122, 11eqtri 2455 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
13 nfcv 2571 . . . . 5  |-  F/_ x A
14 nfopab1 4266 . . . . . . . 8  |-  F/_ x { <. x ,  y
>.  |  ph }
155, 14nfcxfr 2568 . . . . . . 7  |-  F/_ x F
16 nfcv 2571 . . . . . . 7  |-  F/_ x
y
1713, 15, 16nfbr 4248 . . . . . 6  |-  F/ x  A F y
18 nfv 1629 . . . . . 6  |-  F/ x ps
1917, 18nfbi 1856 . . . . 5  |-  F/ x
( A F y  <->  ps )
20 breq1 4207 . . . . . 6  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
21 fvopab5.2 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2220, 21bibi12d 313 . . . . 5  |-  ( x  =  A  ->  (
( x F y  <->  ph )  <->  ( A F y  <->  ps ) ) )
23 df-br 4205 . . . . . 6  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
245eleq2i 2499 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  <->  <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } )
25 opabid 4453 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
2623, 24, 253bitri 263 . . . . 5  |-  ( x F y  <->  ph )
2713, 19, 22, 26vtoclgf 3002 . . . 4  |-  ( A  e.  _V  ->  ( A F y  <->  ps )
)
2827iotabidv 5431 . . 3  |-  ( A  e.  _V  ->  ( iota y A F y )  =  ( iota y ps ) )
2912, 28syl5eq 2479 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota y ps ) )
301, 29syl 16 1  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   {copab 4257   iotacio 5408   ` cfv 5446
This theorem is referenced by:  ajval  22355  adjval  23385
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-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-iota 5410  df-fv 5454
  Copyright terms: Public domain W3C validator