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Theorem qliftfun 6611
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 6607 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 6564 . . 3  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 qliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  B )
81, 5, 2, 6, 7fliftfun 5791 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
93adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  R  Er  X )
10 simpr 110 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  x R y )
119, 10ercl 6540 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  x  e.  X )
129, 10ercl2 6542 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  y  e.  X )
1311, 12jca 306 . . . . . . . . 9  |-  ( (
ph  /\  x R
y )  ->  (
x  e.  X  /\  y  e.  X )
)
1413ex 115 . . . . . . . 8  |-  ( ph  ->  ( x R y  ->  ( x  e.  X  /\  y  e.  X ) ) )
1514pm4.71rd 394 . . . . . . 7  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  x R y ) ) )
163adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  R  Er  X )
17 simprl 529 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
1816, 17erth 6573 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x R y  <->  [ x ] R  =  [ y ] R
) )
1918pm5.32da 452 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  X  /\  y  e.  X )  /\  x R y )  <->  ( (
x  e.  X  /\  y  e.  X )  /\  [ x ] R  =  [ y ] R
) ) )
2015, 19bitrd 188 . . . . . 6  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  [
x ] R  =  [ y ] R
) ) )
2120imbi1d 231 . . . . 5  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
( x  e.  X  /\  y  e.  X
)  /\  [ x ] R  =  [
y ] R )  ->  A  =  B ) ) )
22 impexp 263 . . . . 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, 22bitrdi 196 . . . 4  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
x  e.  X  /\  y  e.  X )  ->  ( [ x ] R  =  [ y ] R  ->  A  =  B ) ) ) )
24232albidv 1867 . . 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 2496 . . 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, 25bitr4di 198 . 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 191 1  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737   <.cop 3594   class class class wbr 4000    |-> cmpt 4061   ran crn 4624   Fun wfun 5206    Er wer 6526   [cec 6527   /.cqs 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-er 6529  df-ec 6531  df-qs 6535
This theorem is referenced by:  qliftfund  6612  qliftfuns  6613
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