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Theorem fvmptex 5772
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 5689.) 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 3279 . . . 4  |-  ( y  =  C  ->  [_ y  /  x ]_ B  = 
[_ C  /  x ]_ B )
2 fvmptex.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
3 nfcv 2569 . . . . . 6  |-  F/_ y B
4 nfcsb1v 3292 . . . . . 6  |-  F/_ x [_ y  /  x ]_ B
5 csbeq1a 3285 . . . . . 6  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
63, 4, 5cbvmpt 4370 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  [_ y  /  x ]_ B )
72, 6eqtri 2453 . . . 4  |-  F  =  ( y  e.  A  |-> 
[_ y  /  x ]_ B )
81, 7fvmpti 5761 . . 3  |-  ( C  e.  A  ->  ( F `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
91fveq2d 5683 . . . 4  |-  ( y  =  C  ->  (  _I  `  [_ y  /  x ]_ B )  =  (  _I  `  [_ C  /  x ]_ B ) )
10 fvmptex.2 . . . . 5  |-  G  =  ( x  e.  A  |->  (  _I  `  B
) )
11 nfcv 2569 . . . . . 6  |-  F/_ y
(  _I  `  B
)
12 nfcv 2569 . . . . . . 7  |-  F/_ x  _I
1312, 4nffv 5686 . . . . . 6  |-  F/_ x
(  _I  `  [_ y  /  x ]_ B )
145fveq2d 5683 . . . . . 6  |-  ( x  =  y  ->  (  _I  `  B )  =  (  _I  `  [_ y  /  x ]_ B ) )
1511, 13, 14cbvmpt 4370 . . . . 5  |-  ( x  e.  A  |->  (  _I 
`  B ) )  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
1610, 15eqtri 2453 . . . 4  |-  G  =  ( y  e.  A  |->  (  _I  `  [_ y  /  x ]_ B ) )
17 fvex 5689 . . . 4  |-  (  _I 
`  [_ C  /  x ]_ B )  e.  _V
189, 16, 17fvmpt 5762 . . 3  |-  ( C  e.  A  ->  ( G `  C )  =  (  _I  `  [_ C  /  x ]_ B ) )
198, 18eqtr4d 2468 . 2  |-  ( C  e.  A  ->  ( F `  C )  =  ( G `  C ) )
202dmmptss 5322 . . . . . 6  |-  dom  F  C_  A
2120sseli 3340 . . . . 5  |-  ( C  e.  dom  F  ->  C  e.  A )
2221con3i 135 . . . 4  |-  ( -.  C  e.  A  ->  -.  C  e.  dom  F )
23 ndmfv 5702 . . . 4  |-  ( -.  C  e.  dom  F  ->  ( F `  C
)  =  (/) )
2422, 23syl 16 . . 3  |-  ( -.  C  e.  A  -> 
( F `  C
)  =  (/) )
25 fvex 5689 . . . . . 6  |-  (  _I 
`  B )  e. 
_V
2625, 10dmmpti 5528 . . . . 5  |-  dom  G  =  A
2726eleq2i 2497 . . . 4  |-  ( C  e.  dom  G  <->  C  e.  A )
28 ndmfv 5702 . . . 4  |-  ( -.  C  e.  dom  G  ->  ( G `  C
)  =  (/) )
2927, 28sylnbir 307 . . 3  |-  ( -.  C  e.  A  -> 
( G `  C
)  =  (/) )
3024, 29eqtr4d 2468 . 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 1362    e. wcel 1755   [_csb 3276   (/)c0 3625    e. cmpt 4338    _I cid 4618   dom cdm 4827   ` cfv 5406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414
This theorem is referenced by:  fvmptnf  5779  sumeq2ii  13153  prodeq2ii  27272
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