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Theorem fvopab5 6289
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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 df-fv 5263 . . . 4  |-  ( F `
 A )  =  ( iota z A F z )
3 breq2 4027 . . . . 5  |-  ( z  =  y  ->  ( A F z  <->  A F
y ) )
4 nfcv 2419 . . . . . 6  |-  F/_ y A
5 fvopab5.1 . . . . . . 7  |-  F  =  { <. x ,  y
>.  |  ph }
6 nfopab2 4086 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
75, 6nfcxfr 2416 . . . . . 6  |-  F/_ y F
8 nfcv 2419 . . . . . 6  |-  F/_ y
z
94, 7, 8nfbr 4067 . . . . 5  |-  F/ y  A F z
10 nfv 1605 . . . . 5  |-  F/ z  A F y
113, 9, 10cbviota 5224 . . . 4  |-  ( iota z A F z )  =  ( iota y A F y )
122, 11eqtri 2303 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
13 nfcv 2419 . . . . 5  |-  F/_ x A
14 nfopab1 4085 . . . . . . . 8  |-  F/_ x { <. x ,  y
>.  |  ph }
155, 14nfcxfr 2416 . . . . . . 7  |-  F/_ x F
16 nfcv 2419 . . . . . . 7  |-  F/_ x
y
1713, 15, 16nfbr 4067 . . . . . 6  |-  F/ x  A F y
18 nfv 1605 . . . . . 6  |-  F/ x ps
1917, 18nfbi 1772 . . . . 5  |-  F/ x
( A F y  <->  ps )
20 breq1 4026 . . . . . 6  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
21 fvopab5.2 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2220, 21bibi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( x F y  <->  ph )  <->  ( A F y  <->  ps ) ) )
23 df-br 4024 . . . . . 6  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
245eleq2i 2347 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  <->  <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } )
25 opabid 4271 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
2623, 24, 253bitri 262 . . . . 5  |-  ( x F y  <->  ph )
2713, 19, 22, 26vtoclgf 2842 . . . 4  |-  ( A  e.  _V  ->  ( A F y  <->  ps )
)
2827iotabidv 5240 . . 3  |-  ( A  e.  _V  ->  ( iota y A F y )  =  ( iota y ps ) )
2912, 28syl5eq 2327 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota y ps ) )
301, 29syl 15 1  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   {copab 4076   iotacio 5217   ` cfv 5255
This theorem is referenced by:  ajval  21440  adjval  22470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263
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