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

Theorem fvmptex 5868
Description: Express a function  F whose value  B may not always be a set in terms of another function  G for which sethood is guaranteed. (Note that  (  _I  `  B ) is just shorthand for  if ( B  e.  _V ,  B ,  (/) ), and it is always a set by fvex 5784.) Note also that these functions are not the same; wherever  B
( C ) is not a set,  C is not in the domain of  F (so it evaluates to the empty set), but  C is in the domain of  G, and  G ( C ) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fvmptex.1  |-  F  =  ( x  e.  A  |->  B )
fvmptex.2  |-  G  =  ( x  e.  A  |->  (  _I  `  B
) )
Assertion
Ref Expression
fvmptex  |-  ( F `
 C )  =  ( G `  C
)
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    F( x)    G( x)

Proof of Theorem fvmptex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3351 . . . 4  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmptex.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2544 . . . . . 6  |-  F/_ y B
4 nfcsb1v 3364 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3357 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4457 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2411 . . . 4  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmpti 5856 . . 3  |-  ( C  e.  A  ->  ( F `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
91fveq2d 5778 . . . 4  |-  ( y  =  C  ->  (  _I  `  [_ y  /  x ]_ B )  =  (  _I  `  [_ C  /  x ]_ B ) )
10 fvmptex.2 . . . . 5  |-  G  =  ( x  e.  A  |->  (  _I  `  B
) )
11 nfcv 2544 . . . . . 6  |-  F/_ y
(  _I  `  B
)
12 nfcv 2544 . . . . . . 7  |-  F/_ x  _I
1312, 4nffv 5781 . . . . . 6  |-  F/_ x
(  _I  `  [_ y  /  x ]_ B )
145fveq2d 5778 . . . . . 6  |-  ( x  =  y  ->  (  _I  `  B )  =  (  _I  `  [_ y  /  x ]_ B ) )
1511, 13, 14cbvmpt 4457 . . . . 5  |-  ( x  e.  A  |->  (  _I 
`  B ) )  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
1610, 15eqtri 2411 . . . 4  |-  G  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
17 fvex 5784 . . . 4  |-  (  _I 
`  [_ C  /  x ]_ B )  e.  _V
189, 16, 17fvmpt 5857 . . 3  |-  ( C  e.  A  ->  ( G `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
198, 18eqtr4d 2426 . 2  |-  ( C  e.  A  ->  ( F `  C )  =  ( G `  C ) )
202dmmptss 5411 . . . . . 6  |-  dom  F  C_  A
2120sseli 3413 . . . . 5  |-  ( C  e.  dom  F  ->  C  e.  A )
2221con3i 135 . . . 4  |-  ( -.  C  e.  A  ->  -.  C  e.  dom  F )
23 ndmfv 5798 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
2422, 23syl 16 . . 3  |-  ( -.  C  e.  A  -> 
( F `  C
)  =  (/) )
25 fvex 5784 . . . . . 6  |-  (  _I 
`  B )  e. 
_V
2625, 10dmmpti 5618 . . . . 5  |-  dom  G  =  A
2726eleq2i 2460 . . . 4  |-  ( C  e.  dom  G  <->  C  e.  A )
28 ndmfv 5798 . . . 4  |-  ( -.  C  e.  dom  G  ->  ( G `  C
)  =  (/) )
2927, 28sylnbir 305 . . 3  |-  ( -.  C  e.  A  -> 
( G `  C
)  =  (/) )
3024, 29eqtr4d 2426 . 2  |-  ( -.  C  e.  A  -> 
( F `  C
)  =  ( G `
 C ) )
3119, 30pm2.61i 164 1  |-  ( F `
 C )  =  ( G `  C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1399    e. wcel 1826   [_csb 3348   (/)c0 3711    |-> cmpt 4425    _I cid 4704   dom cdm 4913   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504
This theorem is referenced by:  fvmptnf  5875  sumeq2ii  13517  prodeq2ii  13722
  Copyright terms: Public domain W3C validator