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Theorem fliftval 5579
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 2089 . . . . 5  |-  ( ph  ->  D  =  D )
5 eqidd 2089 . . . . 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 2099 . . . . . 6  |-  ( x  =  Y  ->  ( C  =  A  <->  C  =  C ) )
9 fliftval.5 . . . . . . 7  |-  ( x  =  Y  ->  B  =  D )
109eqeq2d 2099 . . . . . 6  |-  ( x  =  Y  ->  ( D  =  B  <->  D  =  D ) )
118, 10anbi12d 457 . . . . 5  |-  ( x  =  Y  ->  (
( C  =  A  /\  D  =  B )  <->  ( C  =  C  /\  D  =  D ) ) )
1211rspcev 2722 . . . 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 5572 . . . 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 5343 . 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 1289    e. wcel 1438   E.wrex 2360   <.cop 3449   class class class wbr 3845    |-> cmpt 3899   ran crn 4439   Fun wfun 5009   ` cfv 5015
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by:  qliftval  6376
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