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Theorem fliftfund 6221
Description: The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
fliftfun.4  |-  ( x  =  y  ->  A  =  C )
fliftfun.5  |-  ( x  =  y  ->  B  =  D )
fliftfund.6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  A  =  C ) )  ->  B  =  D )
Assertion
Ref Expression
fliftfund  |-  ( ph  ->  Fun  F )
Distinct variable groups:    y, A    y, B    x, C    x, y, R    x, D    y, F    ph, x, y    x, X, y    x, S, y
Allowed substitution hints:    A( x)    B( x)    C( y)    D( y)    F( x)

Proof of Theorem fliftfund
StepHypRef Expression
1 fliftfund.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  A  =  C ) )  ->  B  =  D )
213exp2 1223 . . . 4  |-  ( ph  ->  ( x  e.  X  ->  ( y  e.  X  ->  ( A  =  C  ->  B  =  D ) ) ) )
32imp32 434 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( A  =  C  ->  B  =  D ) )
43ralrimivva 2853 . 2  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) )
5 flift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
6 flift.2 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
7 flift.3 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
8 fliftfun.4 . . 3  |-  ( x  =  y  ->  A  =  C )
9 fliftfun.5 . . 3  |-  ( x  =  y  ->  B  =  D )
105, 6, 7, 8, 9fliftfun 6220 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
114, 10mpbird 235 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   <.cop 4008    |-> cmpt 4484   ran crn 4855   Fun wfun 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609
This theorem is referenced by:  cygznlem2a  19069  pi1xfrf  21977  pi1cof  21983
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