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Theorem fliftval 6189
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 463 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  Fun  F )
3 simpr 459 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  Y  e.  X )
4 eqidd 2455 . . . . 5  |-  ( ph  ->  D  =  D )
5 eqidd 2455 . . . . 5  |-  ( Y  e.  X  ->  C  =  C )
64, 5anim12ci 565 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  ( C  =  C  /\  D  =  D )
)
7 fliftval.4 . . . . . . 7  |-  ( x  =  Y  ->  A  =  C )
87eqeq2d 2468 . . . . . 6  |-  ( x  =  Y  ->  ( C  =  A  <->  C  =  C ) )
9 fliftval.5 . . . . . . 7  |-  ( x  =  Y  ->  B  =  D )
109eqeq2d 2468 . . . . . 6  |-  ( x  =  Y  ->  ( D  =  B  <->  D  =  D ) )
118, 10anbi12d 708 . . . . 5  |-  ( x  =  Y  ->  (
( C  =  A  /\  D  =  B )  <->  ( C  =  C  /\  D  =  D ) ) )
1211rspcev 3207 . . . 4  |-  ( ( Y  e.  X  /\  ( C  =  C  /\  D  =  D
) )  ->  E. x  e.  X  ( C  =  A  /\  D  =  B ) )
133, 6, 12syl2anc 659 . . 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 6182 . . . 4  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1817adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  X )  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
1913, 18mpbird 232 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  C F D )
20 funbrfv 5886 . 2  |-  ( Fun 
F  ->  ( C F D  ->  ( F `
 C )  =  D ) )
212, 19, 20sylc 60 1  |-  ( (
ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   Fun wfun 5564   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  qliftval  7392  cygznlem2  18780  pi1xfrval  21720  pi1coval  21726
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