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Theorem qliftfun 6981
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 6977 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 6933 . . 3  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 qliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  B )
81, 5, 2, 6, 7fliftfun 6026 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
93adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  R  Er  X )
10 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  x R y )
119, 10ercl 6908 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  x  e.  X )
129, 10ercl2 6910 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  y  e.  X )
1311, 12jca 519 . . . . . . . . 9  |-  ( (
ph  /\  x R
y )  ->  (
x  e.  X  /\  y  e.  X )
)
1413ex 424 . . . . . . . 8  |-  ( ph  ->  ( x R y  ->  ( x  e.  X  /\  y  e.  X ) ) )
1514pm4.71rd 617 . . . . . . 7  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  x R y ) ) )
163adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  R  Er  X )
17 simprl 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
1816, 17erth 6941 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x R y  <->  [ x ] R  =  [ y ] R
) )
1918pm5.32da 623 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  X  /\  y  e.  X )  /\  x R y )  <->  ( (
x  e.  X  /\  y  e.  X )  /\  [ x ] R  =  [ y ] R
) ) )
2015, 19bitrd 245 . . . . . 6  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  [
x ] R  =  [ y ] R
) ) )
2120imbi1d 309 . . . . 5  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
( x  e.  X  /\  y  e.  X
)  /\  [ x ] R  =  [
y ] R )  ->  A  =  B ) ) )
22 impexp 434 . . . . 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 253 . . . 4  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
x  e.  X  /\  y  e.  X )  ->  ( [ x ] R  =  [ y ] R  ->  A  =  B ) ) ) )
24232albidv 1637 . . 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 2734 . . 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 255 . 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 248 1  |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   <.cop 3809   class class class wbr 4204    e. cmpt 4258   ran crn 4871   Fun wfun 5440    Er wer 6894   [cec 6895   /.cqs 6896
This theorem is referenced by:  qliftfund  6982  qliftfuns  6983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-er 6897  df-ec 6899  df-qs 6903
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