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Theorem fliftfund 5828
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 1169 . . . 4  |-  ( ph  ->  ( x  e.  X  ->  ( y  e.  X  ->  ( A  =  C  ->  B  =  D ) ) ) )
32imp32 422 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( A  =  C  ->  B  =  D ) )
43ralrimivva 2648 . 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 5827 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
114, 10mpbird 223 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656    e. cmpt 4093   ran crn 4706   Fun wfun 5265
This theorem is referenced by:  cygznlem2a  16537  pi1xfrf  18567  pi1cof  18573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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