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Theorem divsfval 13465
Description: Value of the function in divsval 13460. (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 6717 . . . . 5  |-  ( ph  ->  [ A ]  .~  C_  V )
3 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
4 ssexg 4176 . . . . 5  |-  ( ( [ A ]  .~  C_  V  /\  V  e. 
_V )  ->  [ A ]  .~  e.  _V )
52, 3, 4syl2anc 642 . . . 4  |-  ( ph  ->  [ A ]  .~  e.  _V )
6 eceq1 6712 . . . . 5  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
7 ercpbl.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
86, 7fvmptg 5616 . . . 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 4896 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  V  |->  [ x ]  .~  )
121ecss 6717 . . . . . . . . . . 11  |-  ( ph  ->  [ x ]  .~  C_  V )
13 ssexg 4176 . . . . . . . . . . 11  |-  ( ( [ x ]  .~  C_  V  /\  V  e. 
_V )  ->  [ x ]  .~  e.  _V )
1412, 3, 13syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  [ x ]  .~  e.  _V )
1514ralrimivw 2640 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
16 dmmptg 5186 . . . . . . . . 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 2340 . . . . . . 7  |-  ( ph  ->  dom  F  =  V )
1918eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  V ) )
2019notbid 285 . . . . 5  |-  ( ph  ->  ( -.  A  e. 
dom  F  <->  -.  A  e.  V ) )
21 ndmfv 5568 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2220, 21syl6bir 220 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  (/) ) )
23 ecdmn0 6718 . . . . . 6  |-  ( A  e.  dom  .~  <->  [ A ]  .~  =/=  (/) )
24 erdm 6686 . . . . . . . . 9  |-  (  .~  Er  V  ->  dom  .~  =  V )
251, 24syl 15 . . . . . . . 8  |-  ( ph  ->  dom  .~  =  V )
2625eleq2d 2363 . . . . . . 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 2527 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  [ A ]  .~  =  (/) ) )
3022, 29jcad 519 . . 3  |-  ( ph  ->  ( -.  A  e.  V  ->  ( ( F `  A )  =  (/)  /\  [ A ]  .~  =  (/) ) ) )
31 eqtr3 2315 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468    e. cmpt 4093   dom cdm 4705   ` cfv 5271    Er wer 6673   [cec 6674
This theorem is referenced by:  ercpbllem  13466  divsaddvallem  13469  divsgrp2  14629  frgpmhm  15090  frgpup3lem  15102  divsrng2  15419  divsrhm  16005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-er 6676  df-ec 6678
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