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Theorem imaiinfv 30229
Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
imaiinfv  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( F `  x )  =  |^| ( F
" B ) )
Distinct variable groups:    x, B    x, F
Allowed substitution hint:    A( x)

Proof of Theorem imaiinfv
StepHypRef Expression
1 fnssres 5692 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fniinfv 5924 . . 3  |-  ( ( F  |`  B )  Fn  B  ->  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^| ran  ( F  |`  B ) )
31, 2syl 16 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( ( F  |`  B ) `
 x )  = 
|^| ran  ( F  |`  B ) )
4 fvres 5878 . . . 4  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
54iineq2i 4345 . . 3  |-  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^|_ x  e.  B  ( F `
 x )
65eqcomi 2480 . 2  |-  |^|_ x  e.  B  ( F `  x )  =  |^|_ x  e.  B  ( ( F  |`  B ) `  x )
7 df-ima 5012 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
87inteqi 4286 . 2  |-  |^| ( F " B )  = 
|^| ran  ( F  |`  B )
93, 6, 83eqtr4g 2533 1  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( F `  x )  =  |^| ( F
" B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    C_ wss 3476   |^|cint 4282   |^|_ciin 4326   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  elrfirn  30231
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