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Theorem fliftval 5471
Description: The value of the function  F. (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 )
fliftval.4  |-  ( x  =  Y  ->  A  =  C )
fliftval.5  |-  ( x  =  Y  ->  B  =  D )
fliftval.6  |-  ( ph  ->  Fun  F )
Assertion
Ref Expression
fliftval  |-  ( (
ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
Distinct variable groups:    x, C    x, R    x, Y    x, D    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftval
StepHypRef Expression
1 fliftval.6 . . 3  |-  ( ph  ->  Fun  F )
21adantr 270 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  Fun  F )
3 simpr 108 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  Y  e.  X )
4 eqidd 2083 . . . . 5  |-  ( ph  ->  D  =  D )
5 eqidd 2083 . . . . 5  |-  ( Y  e.  X  ->  C  =  C )
64, 5anim12ci 332 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  ( C  =  C  /\  D  =  D )
)
7 fliftval.4 . . . . . . 7  |-  ( x  =  Y  ->  A  =  C )
87eqeq2d 2093 . . . . . 6  |-  ( x  =  Y  ->  ( C  =  A  <->  C  =  C ) )
9 fliftval.5 . . . . . . 7  |-  ( x  =  Y  ->  B  =  D )
109eqeq2d 2093 . . . . . 6  |-  ( x  =  Y  ->  ( D  =  B  <->  D  =  D ) )
118, 10anbi12d 457 . . . . 5  |-  ( x  =  Y  ->  (
( C  =  A  /\  D  =  B )  <->  ( C  =  C  /\  D  =  D ) ) )
1211rspcev 2702 . . . 4  |-  ( ( Y  e.  X  /\  ( C  =  C  /\  D  =  D
) )  ->  E. x  e.  X  ( C  =  A  /\  D  =  B ) )
133, 6, 12syl2anc 403 . . 3  |-  ( (
ph  /\  Y  e.  X )  ->  E. x  e.  X  ( C  =  A  /\  D  =  B ) )
14 flift.1 . . . . 5  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
15 flift.2 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
16 flift.3 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
1714, 15, 16fliftel 5464 . . . 4  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1817adantr 270 . . 3  |-  ( (
ph  /\  Y  e.  X )  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
1913, 18mpbird 165 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  C F D )
20 funbrfv 5244 . 2  |-  ( Fun 
F  ->  ( C F D  ->  ( F `
 C )  =  D ) )
212, 19, 20sylc 61 1  |-  ( (
ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350   <.cop 3409   class class class wbr 3793    |-> cmpt 3847   ran crn 4372   Fun wfun 4926   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by:  qliftval  6258
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