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Theorem divsfval 13764
Description: Value of the function in divsval 13759. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
Assertion
Ref Expression
divsfval  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Distinct variable groups:    x,  .~    x, A    x, V    ph, x
Allowed substitution hint:    F( x)

Proof of Theorem divsfval
StepHypRef Expression
1 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
2 ercpbl.r . . . . . 6  |-  ( ph  ->  .~  Er  V )
32ecss 6938 . . . . 5  |-  ( ph  ->  [ A ]  .~  C_  V )
41, 3ssexd 4342 . . . 4  |-  ( ph  ->  [ A ]  .~  e.  _V )
5 eceq1 6933 . . . . 5  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
6 ercpbl.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
75, 6fvmptg 5796 . . . 4  |-  ( ( A  e.  V  /\  [ A ]  .~  e.  _V )  ->  ( F `
 A )  =  [ A ]  .~  )
84, 7sylan2 461 . . 3  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [ A ]  .~  )
98expcom 425 . 2  |-  ( ph  ->  ( A  e.  V  ->  ( F `  A
)  =  [ A ]  .~  ) )
106dmeqi 5063 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  V  |->  [ x ]  .~  )
112ecss 6938 . . . . . . . . . . 11  |-  ( ph  ->  [ x ]  .~  C_  V )
121, 11ssexd 4342 . . . . . . . . . 10  |-  ( ph  ->  [ x ]  .~  e.  _V )
1312ralrimivw 2782 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
14 dmmptg 5359 . . . . . . . . 9  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V
)
1610, 15syl5eq 2479 . . . . . . 7  |-  ( ph  ->  dom  F  =  V )
1716eleq2d 2502 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  V ) )
1817notbid 286 . . . . 5  |-  ( ph  ->  ( -.  A  e. 
dom  F  <->  -.  A  e.  V ) )
19 ndmfv 5747 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2018, 19syl6bir 221 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  (/) ) )
21 ecdmn0 6939 . . . . . 6  |-  ( A  e.  dom  .~  <->  [ A ]  .~  =/=  (/) )
22 erdm 6907 . . . . . . . . 9  |-  (  .~  Er  V  ->  dom  .~  =  V )
232, 22syl 16 . . . . . . . 8  |-  ( ph  ->  dom  .~  =  V )
2423eleq2d 2502 . . . . . . 7  |-  ( ph  ->  ( A  e.  dom  .~  <->  A  e.  V ) )
2524biimpd 199 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  .~ 
->  A  e.  V
) )
2621, 25syl5bir 210 . . . . 5  |-  ( ph  ->  ( [ A ]  .~  =/=  (/)  ->  A  e.  V ) )
2726necon1bd 2666 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  [ A ]  .~  =  (/) ) )
2820, 27jcad 520 . . 3  |-  ( ph  ->  ( -.  A  e.  V  ->  ( ( F `  A )  =  (/)  /\  [ A ]  .~  =  (/) ) ) )
29 eqtr3 2454 . . 3  |-  ( ( ( F `  A
)  =  (/)  /\  [ A ]  .~  =  (/) )  ->  ( F `  A )  =  [ A ]  .~  )
3028, 29syl6 31 . 2  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  [ A ]  .~  )
)
319, 30pm2.61d 152 1  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948   (/)c0 3620    e. cmpt 4258   dom cdm 4870   ` cfv 5446    Er wer 6894   [cec 6895
This theorem is referenced by:  ercpbllem  13765  divsaddvallem  13768  divsgrp2  14928  frgpmhm  15389  frgpup3lem  15401  divsrng2  15718  divsrhm  16300
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-er 6897  df-ec 6899
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