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Theorem fliftval 6030
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 465 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  Fun  F )
3 simpr 461 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  Y  e.  X )
4 eqidd 2444 . . . . 5  |-  ( ph  ->  D  =  D )
5 eqidd 2444 . . . . 5  |-  ( Y  e.  X  ->  C  =  C )
64, 5anim12ci 567 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  ( C  =  C  /\  D  =  D )
)
7 fliftval.4 . . . . . . 7  |-  ( x  =  Y  ->  A  =  C )
87eqeq2d 2454 . . . . . 6  |-  ( x  =  Y  ->  ( C  =  A  <->  C  =  C ) )
9 fliftval.5 . . . . . . 7  |-  ( x  =  Y  ->  B  =  D )
109eqeq2d 2454 . . . . . 6  |-  ( x  =  Y  ->  ( D  =  B  <->  D  =  D ) )
118, 10anbi12d 710 . . . . 5  |-  ( x  =  Y  ->  (
( C  =  A  /\  D  =  B )  <->  ( C  =  C  /\  D  =  D ) ) )
1211rspcev 3094 . . . 4  |-  ( ( Y  e.  X  /\  ( C  =  C  /\  D  =  D
) )  ->  E. x  e.  X  ( C  =  A  /\  D  =  B ) )
133, 6, 12syl2anc 661 . . 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 6023 . . . 4  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1817adantr 465 . . 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 5751 . 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 369    = wceq 1369    e. wcel 1756   E.wrex 2737   <.cop 3904   class class class wbr 4313    e. cmpt 4371   ran crn 4862   Fun wfun 5433   ` cfv 5439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447
This theorem is referenced by:  qliftval  7210  cygznlem2  18023  pi1xfrval  20648  pi1coval  20654
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