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Theorem fliftfund 5666
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 1188 . . . 4  |-  ( ph  ->  ( x  e.  X  ->  ( y  e.  X  ->  ( A  =  C  ->  B  =  D ) ) ) )
32imp32 255 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( A  =  C  ->  B  =  D ) )
43ralrimivva 2491 . 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 5665 . 2  |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
114, 10mpbird 166 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   <.cop 3500    |-> cmpt 3959   ran crn 4510   Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by: (None)
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