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Theorem fvmptex 5610
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 5539.) 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 3084 . . . 4  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmptex.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2419 . . . . . 6  |-  F/_ y B
4 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3089 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4110 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2303 . . . 4  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmpti 5601 . . 3  |-  ( C  e.  A  ->  ( F `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
91fveq2d 5529 . . . 4  |-  ( y  =  C  ->  (  _I  `  [_ y  /  x ]_ B )  =  (  _I  `  [_ C  /  x ]_ B ) )
10 fvmptex.2 . . . . 5  |-  G  =  ( x  e.  A  |->  (  _I  `  B
) )
11 nfcv 2419 . . . . . 6  |-  F/_ y
(  _I  `  B
)
12 nfcv 2419 . . . . . . 7  |-  F/_ x  _I
1312, 4nffv 5532 . . . . . 6  |-  F/_ x
(  _I  `  [_ y  /  x ]_ B )
145fveq2d 5529 . . . . . 6  |-  ( x  =  y  ->  (  _I  `  B )  =  (  _I  `  [_ y  /  x ]_ B ) )
1511, 13, 14cbvmpt 4110 . . . . 5  |-  ( x  e.  A  |->  (  _I 
`  B ) )  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
1610, 15eqtri 2303 . . . 4  |-  G  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
17 fvex 5539 . . . 4  |-  (  _I 
`  [_ C  /  x ]_ B )  e.  _V
189, 16, 17fvmpt 5602 . . 3  |-  ( C  e.  A  ->  ( G `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
198, 18eqtr4d 2318 . 2  |-  ( C  e.  A  ->  ( F `  C )  =  ( G `  C ) )
202dmmptss 5169 . . . . . 6  |-  dom  F  C_  A
2120sseli 3176 . . . . 5  |-  ( C  e.  dom  F  ->  C  e.  A )
2221con3i 127 . . . 4  |-  ( -.  C  e.  A  ->  -.  C  e.  dom  F )
23 ndmfv 5552 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
2422, 23syl 15 . . 3  |-  ( -.  C  e.  A  -> 
( F `  C
)  =  (/) )
25 fvex 5539 . . . . . 6  |-  (  _I 
`  B )  e. 
_V
2625, 10dmmpti 5373 . . . . 5  |-  dom  G  =  A
2726eleq2i 2347 . . . 4  |-  ( C  e.  dom  G  <->  C  e.  A )
28 ndmfv 5552 . . . 4  |-  ( -.  C  e.  dom  G  ->  ( G `  C
)  =  (/) )
2927, 28sylnbir 298 . . 3  |-  ( -.  C  e.  A  -> 
( G `  C
)  =  (/) )
3024, 29eqtr4d 2318 . 2  |-  ( -.  C  e.  A  -> 
( F `  C
)  =  ( G `
 C ) )
3119, 30pm2.61i 156 1  |-  ( F `
 C )  =  ( G `  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   [_csb 3081   (/)c0 3455    e. cmpt 4077    _I cid 4304   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  fvmptnf  5617  sumeq2ii  12166
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-csb 3082  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-fn 5258  df-fv 5263
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