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Theorem fvopab5 5971
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 3122 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 df-fv 5594 . . . 4  |-  ( F `
 A )  =  ( iota z A F z )
3 breq2 4451 . . . . 5  |-  ( z  =  y  ->  ( A F z  <->  A F
y ) )
4 nfcv 2629 . . . . . 6  |-  F/_ y A
5 fvopab5.1 . . . . . . 7  |-  F  =  { <. x ,  y
>.  |  ph }
6 nfopab2 4514 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
75, 6nfcxfr 2627 . . . . . 6  |-  F/_ y F
8 nfcv 2629 . . . . . 6  |-  F/_ y
z
94, 7, 8nfbr 4491 . . . . 5  |-  F/ y  A F z
10 nfv 1683 . . . . 5  |-  F/ z  A F y
113, 9, 10cbviota 5554 . . . 4  |-  ( iota z A F z )  =  ( iota y A F y )
122, 11eqtri 2496 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
13 nfcv 2629 . . . . . . 7  |-  F/_ x A
14 nfopab1 4513 . . . . . . . 8  |-  F/_ x { <. x ,  y
>.  |  ph }
155, 14nfcxfr 2627 . . . . . . 7  |-  F/_ x F
16 nfcv 2629 . . . . . . 7  |-  F/_ x
y
1713, 15, 16nfbr 4491 . . . . . 6  |-  F/ x  A F y
18 nfv 1683 . . . . . 6  |-  F/ x ps
1917, 18nfbi 1881 . . . . 5  |-  F/ x
( A F y  <->  ps )
20 breq1 4450 . . . . . 6  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
21 fvopab5.2 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2220, 21bibi12d 321 . . . . 5  |-  ( x  =  A  ->  (
( x F y  <->  ph )  <->  ( A F y  <->  ps ) ) )
23 df-br 4448 . . . . . 6  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
245eleq2i 2545 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  <->  <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } )
25 opabid 4754 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
2623, 24, 253bitri 271 . . . . 5  |-  ( x F y  <->  ph )
2719, 22, 26vtoclg1f 3170 . . . 4  |-  ( A  e.  _V  ->  ( A F y  <->  ps )
)
2827iotabidv 5570 . . 3  |-  ( A  e.  _V  ->  ( iota y A F y )  =  ( iota y ps ) )
2912, 28syl5eq 2520 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447   {copab 4504   iotacio 5547   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5549  df-fv 5594
This theorem is referenced by:  ajval  25453  adjval  26485
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