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Theorem fliftfund 6035
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 1171 . . . 4  |-  ( ph  ->  ( x  e.  X  ->  ( y  e.  X  ->  ( A  =  C  ->  B  =  D ) ) ) )
32imp32 423 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( A  =  C  ->  B  =  D ) )
43ralrimivva 2798 . 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 6034 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
114, 10mpbird 224 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817    e. cmpt 4266   ran crn 4879   Fun wfun 5448
This theorem is referenced by:  cygznlem2a  16848  pi1xfrf  19078  pi1cof  19084
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462
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