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Theorem fliftfund 5812
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 2635 . 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 5811 . 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 1623    e. wcel 1684   A.wral 2543   <.cop 3643    e. cmpt 4077   ran crn 4690   Fun wfun 5249
This theorem is referenced by:  cygznlem2a  16521  pi1xfrf  18551  pi1cof  18557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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