MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftval Structured version   Unicode version

Theorem fliftval 6038
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 452 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  Fun  F )
3 simpr 448 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  Y  e.  X )
4 eqidd 2437 . . . . 5  |-  ( ph  ->  D  =  D )
5 eqidd 2437 . . . . 5  |-  ( Y  e.  X  ->  C  =  C )
64, 5anim12ci 551 . . . 4  |-  ( (
ph  /\  Y  e.  X )  ->  ( C  =  C  /\  D  =  D )
)
7 fliftval.4 . . . . . . 7  |-  ( x  =  Y  ->  A  =  C )
87eqeq2d 2447 . . . . . 6  |-  ( x  =  Y  ->  ( C  =  A  <->  C  =  C ) )
9 fliftval.5 . . . . . . 7  |-  ( x  =  Y  ->  B  =  D )
109eqeq2d 2447 . . . . . 6  |-  ( x  =  Y  ->  ( D  =  B  <->  D  =  D ) )
118, 10anbi12d 692 . . . . 5  |-  ( x  =  Y  ->  (
( C  =  A  /\  D  =  B )  <->  ( C  =  C  /\  D  =  D ) ) )
1211rspcev 3052 . . . 4  |-  ( ( Y  e.  X  /\  ( C  =  C  /\  D  =  D
) )  ->  E. x  e.  X  ( C  =  A  /\  D  =  B ) )
133, 6, 12syl2anc 643 . . 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 6031 . . . 4  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1817adantr 452 . . 3  |-  ( (
ph  /\  Y  e.  X )  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
1913, 18mpbird 224 . 2  |-  ( (
ph  /\  Y  e.  X )  ->  C F D )
20 funbrfv 5765 . 2  |-  ( Fun 
F  ->  ( C F D  ->  ( F `
 C )  =  D ) )
212, 19, 20sylc 58 1  |-  ( (
ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   <.cop 3817   class class class wbr 4212    e. cmpt 4266   ran crn 4879   Fun wfun 5448   ` cfv 5454
This theorem is referenced by:  qliftval  6993  cygznlem2  16849  pi1xfrval  19079  pi1coval  19085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462
  Copyright terms: Public domain W3C validator