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

Theorem divsfval 13449
Description: Value of the function in divsval 13444. (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.r . . . . . 6  |-  ( ph  ->  .~  Er  V )
21ecss 6701 . . . . 5  |-  ( ph  ->  [ A ]  .~  C_  V )
3 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
4 ssexg 4160 . . . . 5  |-  ( ( [ A ]  .~  C_  V  /\  V  e. 
_V )  ->  [ A ]  .~  e.  _V )
52, 3, 4syl2anc 642 . . . 4  |-  ( ph  ->  [ A ]  .~  e.  _V )
6 eceq1 6696 . . . . 5  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
7 ercpbl.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
86, 7fvmptg 5600 . . . 4  |-  ( ( A  e.  V  /\  [ A ]  .~  e.  _V )  ->  ( F `
 A )  =  [ A ]  .~  )
95, 8sylan2 460 . . 3  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [ A ]  .~  )
109expcom 424 . 2  |-  ( ph  ->  ( A  e.  V  ->  ( F `  A
)  =  [ A ]  .~  ) )
117dmeqi 4880 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  V  |->  [ x ]  .~  )
121ecss 6701 . . . . . . . . . . 11  |-  ( ph  ->  [ x ]  .~  C_  V )
13 ssexg 4160 . . . . . . . . . . 11  |-  ( ( [ x ]  .~  C_  V  /\  V  e. 
_V )  ->  [ x ]  .~  e.  _V )
1412, 3, 13syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  [ x ]  .~  e.  _V )
1514ralrimivw 2627 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
16 dmmptg 5170 . . . . . . . . 9  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V )
1715, 16syl 15 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V
)
1811, 17syl5eq 2327 . . . . . . 7  |-  ( ph  ->  dom  F  =  V )
1918eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  V ) )
2019notbid 285 . . . . 5  |-  ( ph  ->  ( -.  A  e. 
dom  F  <->  -.  A  e.  V ) )
21 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2220, 21syl6bir 220 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  (/) ) )
23 ecdmn0 6702 . . . . . 6  |-  ( A  e.  dom  .~  <->  [ A ]  .~  =/=  (/) )
24 erdm 6670 . . . . . . . . 9  |-  (  .~  Er  V  ->  dom  .~  =  V )
251, 24syl 15 . . . . . . . 8  |-  ( ph  ->  dom  .~  =  V )
2625eleq2d 2350 . . . . . . 7  |-  ( ph  ->  ( A  e.  dom  .~  <->  A  e.  V ) )
2726biimpd 198 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  .~ 
->  A  e.  V
) )
2823, 27syl5bir 209 . . . . 5  |-  ( ph  ->  ( [ A ]  .~  =/=  (/)  ->  A  e.  V ) )
2928necon1bd 2514 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  [ A ]  .~  =  (/) ) )
3022, 29jcad 519 . . 3  |-  ( ph  ->  ( -.  A  e.  V  ->  ( ( F `  A )  =  (/)  /\  [ A ]  .~  =  (/) ) ) )
31 eqtr3 2302 . . 3  |-  ( ( ( F `  A
)  =  (/)  /\  [ A ]  .~  =  (/) )  ->  ( F `  A )  =  [ A ]  .~  )
3230, 31syl6 29 . 2  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  [ A ]  .~  )
)
3310, 32pm2.61d 150 1  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455    e. cmpt 4077   dom cdm 4689   ` cfv 5255    Er wer 6657   [cec 6658
This theorem is referenced by:  ercpbllem  13450  divsaddvallem  13453  divsgrp2  14613  frgpmhm  15074  frgpup3lem  15086  divsrng2  15403  divsrhm  15989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-er 6660  df-ec 6662
  Copyright terms: Public domain W3C validator