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

Theorem fvmptf 5982
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5962 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
fvmptf.1  |-  F/_ x A
fvmptf.2  |-  F/_ x C
fvmptf.3  |-  ( x  =  A  ->  B  =  C )
fvmptf.4  |-  F  =  ( x  e.  D  |->  B )
Assertion
Ref Expression
fvmptf  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Distinct variable group:    x, D
Allowed substitution hints:    A( x)    B( x)    C( x)    F( x)    V( x)

Proof of Theorem fvmptf
StepHypRef Expression
1 elex 3096 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
2 fvmptf.1 . . . 4  |-  F/_ x A
3 fvmptf.2 . . . . . 6  |-  F/_ x C
43nfel1 2607 . . . . 5  |-  F/ x  C  e.  _V
5 fvmptf.4 . . . . . . . 8  |-  F  =  ( x  e.  D  |->  B )
6 nfmpt1 4515 . . . . . . . 8  |-  F/_ x
( x  e.  D  |->  B )
75, 6nfcxfr 2589 . . . . . . 7  |-  F/_ x F
87, 2nffv 5888 . . . . . 6  |-  F/_ x
( F `  A
)
98, 3nfeq 2602 . . . . 5  |-  F/ x
( F `  A
)  =  C
104, 9nfim 1978 . . . 4  |-  F/ x
( C  e.  _V  ->  ( F `  A
)  =  C )
11 fvmptf.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
1211eleq1d 2498 . . . . 5  |-  ( x  =  A  ->  ( B  e.  _V  <->  C  e.  _V ) )
13 fveq2 5881 . . . . . 6  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1413, 11eqeq12d 2451 . . . . 5  |-  ( x  =  A  ->  (
( F `  x
)  =  B  <->  ( F `  A )  =  C ) )
1512, 14imbi12d 321 . . . 4  |-  ( x  =  A  ->  (
( B  e.  _V  ->  ( F `  x
)  =  B )  <-> 
( C  e.  _V  ->  ( F `  A
)  =  C ) ) )
165fvmpt2 5973 . . . . 5  |-  ( ( x  e.  D  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
1716ex 435 . . . 4  |-  ( x  e.  D  ->  ( B  e.  _V  ->  ( F `  x )  =  B ) )
182, 10, 15, 17vtoclgaf 3150 . . 3  |-  ( A  e.  D  ->  ( C  e.  _V  ->  ( F `  A )  =  C ) )
191, 18syl5 33 . 2  |-  ( A  e.  D  ->  ( C  e.  V  ->  ( F `  A )  =  C ) )
2019imp 430 1  |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A
)  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   F/_wnfc 2577   _Vcvv 3087    |-> cmpt 4484   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  fvmptnf  5983  elfvmptrab1  5986  elovmpt3rab1  6541  rdgsucmptf  7154  frsucmpt  7163  fprodntriv  13974  prodss  13979  fprodefsum  14127  dvfsumabs  22852  dvfsumlem1  22855  dvfsumlem4  22858  dvfsum2  22863  dchrisumlem2  24191  dchrisumlem3  24192  ptrest  31642  hlhilset  35213  mulc1cncfg  37238  expcnfg  37242  stoweidlem23  37451  stoweidlem34  37463  stoweidlem36  37465  wallispilem5  37499  stirlinglem4  37507  stirlinglem11  37514  stirlinglem12  37515  stirlinglem13  37516  stirlinglem14  37517  sge0lempt  37785  meadjiun  37812
  Copyright terms: Public domain W3C validator