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Theorem qliftfun 7282
Description: The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
qlift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
qlift.3  |-  ( ph  ->  R  Er  X )
qlift.4  |-  ( ph  ->  X  e.  _V )
qliftfun.4  |-  ( x  =  y  ->  A  =  B )
Assertion
Ref Expression
qliftfun  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
Distinct variable groups:    y, A    x, B    x, y, ph    x, R, y    y, F   
x, X, y    x, Y, y
Allowed substitution hints:    A( x)    B( y)    F( x)

Proof of Theorem qliftfun
StepHypRef Expression
1 qlift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
2 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
3 qlift.3 . . . 4  |-  ( ph  ->  R  Er  X )
4 qlift.4 . . . 4  |-  ( ph  ->  X  e.  _V )
51, 2, 3, 4qliftlem 7278 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 7234 . . 3  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 qliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  B )
81, 5, 2, 6, 7fliftfun 6101 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
93adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  R  Er  X )
10 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  x R y )
119, 10ercl 7209 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  x  e.  X )
129, 10ercl2 7211 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  y  e.  X )
1311, 12jca 532 . . . . . . . . 9  |-  ( (
ph  /\  x R
y )  ->  (
x  e.  X  /\  y  e.  X )
)
1413ex 434 . . . . . . . 8  |-  ( ph  ->  ( x R y  ->  ( x  e.  X  /\  y  e.  X ) ) )
1514pm4.71rd 635 . . . . . . 7  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  x R y ) ) )
163adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  R  Er  X )
17 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
1816, 17erth 7242 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x R y  <->  [ x ] R  =  [ y ] R
) )
1918pm5.32da 641 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  X  /\  y  e.  X )  /\  x R y )  <->  ( (
x  e.  X  /\  y  e.  X )  /\  [ x ] R  =  [ y ] R
) ) )
2015, 19bitrd 253 . . . . . 6  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  [
x ] R  =  [ y ] R
) ) )
2120imbi1d 317 . . . . 5  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
( x  e.  X  /\  y  e.  X
)  /\  [ x ] R  =  [
y ] R )  ->  A  =  B ) ) )
22 impexp 446 . . . . 5  |-  ( ( ( ( x  e.  X  /\  y  e.  X )  /\  [
x ] R  =  [ y ] R
)  ->  A  =  B )  <->  ( (
x  e.  X  /\  y  e.  X )  ->  ( [ x ] R  =  [ y ] R  ->  A  =  B ) ) )
2321, 22syl6bb 261 . . . 4  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
x  e.  X  /\  y  e.  X )  ->  ( [ x ] R  =  [ y ] R  ->  A  =  B ) ) ) )
24232albidv 1682 . . 3  |-  ( ph  ->  ( A. x A. y ( x R y  ->  A  =  B )  <->  A. x A. y ( ( x  e.  X  /\  y  e.  X )  ->  ( [ x ] R  =  [ y ] R  ->  A  =  B ) ) ) )
25 r2al 2853 . . 3  |-  ( A. x  e.  X  A. y  e.  X  ( [ x ] R  =  [ y ] R  ->  A  =  B )  <->  A. x A. y ( ( x  e.  X  /\  y  e.  X
)  ->  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
2624, 25syl6bbr 263 . 2  |-  ( ph  ->  ( A. x A. y ( x R y  ->  A  =  B )  <->  A. x  e.  X  A. y  e.  X  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
278, 26bitr4d 256 1  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2793   _Vcvv 3065   <.cop 3978   class class class wbr 4387    |-> cmpt 4445   ran crn 4936   Fun wfun 5507    Er wer 7195   [cec 7196   /.cqs 7197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-er 7198  df-ec 7200  df-qs 7204
This theorem is referenced by:  qliftfund  7283  qliftfuns  7284
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