MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qliftfun Structured version   Unicode version

Theorem qliftfun 7388
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 7384 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 7339 . . 3  |-  ( x  =  y  ->  [ x ] R  =  [
y ] R )
7 qliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  B )
81, 5, 2, 6, 7fliftfun 6185 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( [
x ] R  =  [ y ] R  ->  A  =  B ) ) )
93adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  R  Er  X )
10 simpr 459 . . . . . . . . . . 11  |-  ( (
ph  /\  x R
y )  ->  x R y )
119, 10ercl 7314 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  x  e.  X )
129, 10ercl2 7316 . . . . . . . . . 10  |-  ( (
ph  /\  x R
y )  ->  y  e.  X )
1311, 12jca 530 . . . . . . . . 9  |-  ( (
ph  /\  x R
y )  ->  (
x  e.  X  /\  y  e.  X )
)
1413ex 432 . . . . . . . 8  |-  ( ph  ->  ( x R y  ->  ( x  e.  X  /\  y  e.  X ) ) )
1514pm4.71rd 633 . . . . . . 7  |-  ( ph  ->  ( x R y  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  x R y ) ) )
163adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  R  Er  X )
17 simprl 754 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
1816, 17erth 7348 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x R y  <->  [ x ] R  =  [ y ] R
) )
1918pm5.32da 639 . . . . . . 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 315 . . . . 5  |-  ( ph  ->  ( ( x R y  ->  A  =  B )  <->  ( (
( x  e.  X  /\  y  e.  X
)  /\  [ x ] R  =  [
y ] R )  ->  A  =  B ) ) )
22 impexp 444 . . . . 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 1720 . . 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 2832 . . 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 367   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   <.cop 4022   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   Fun wfun 5564    Er wer 7300   [cec 7301   /.cqs 7302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-er 7303  df-ec 7305  df-qs 7309
This theorem is referenced by:  qliftfund  7389  qliftfuns  7390
  Copyright terms: Public domain W3C validator