ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvopab6 Unicode version

Theorem fvopab6 5590
Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab6.1  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
fvopab6.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
fvopab6.3  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
fvopab6  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Distinct variable groups:    x, A, y    ps, x, y    y, B   
x, C, y
Allowed substitution hints:    ph( x, y)    B( x)    D( x, y)    R( x, y)    F( x, y)

Proof of Theorem fvopab6
StepHypRef Expression
1 elex 2741 . . 3  |-  ( A  e.  D  ->  A  e.  _V )
2 fvopab6.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 fvopab6.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
43eqeq2d 2182 . . . . 5  |-  ( x  =  A  ->  (
y  =  B  <->  y  =  C ) )
52, 4anbi12d 470 . . . 4  |-  ( x  =  A  ->  (
( ph  /\  y  =  B )  <->  ( ps  /\  y  =  C ) ) )
6 iba 298 . . . . 5  |-  ( y  =  C  ->  ( ps 
<->  ( ps  /\  y  =  C ) ) )
76bicomd 140 . . . 4  |-  ( y  =  C  ->  (
( ps  /\  y  =  C )  <->  ps )
)
8 moeq 2905 . . . . . 6  |-  E* y 
y  =  B
98moani 2089 . . . . 5  |-  E* y
( ph  /\  y  =  B )
109a1i 9 . . . 4  |-  ( x  e.  _V  ->  E* y ( ph  /\  y  =  B )
)
11 fvopab6.1 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
12 vex 2733 . . . . . . 7  |-  x  e. 
_V
1312biantrur 301 . . . . . 6  |-  ( (
ph  /\  y  =  B )  <->  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) )
1413opabbii 4054 . . . . 5  |-  { <. x ,  y >.  |  (
ph  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
1511, 14eqtri 2191 . . . 4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
165, 7, 10, 15fvopab3ig 5568 . . 3  |-  ( ( A  e.  _V  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
171, 16sylan 281 . 2  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
18173impia 1195 1  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348   E*wmo 2020    e. wcel 2141   _Vcvv 2730   {copab 4047   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator