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Theorem imaiinfv 29048
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 5539 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fniinfv 5765 . . 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 5719 . . . 4  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
54iineq2i 4205 . . 3  |-  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^|_ x  e.  B  ( F `
 x )
65eqcomi 2447 . 2  |-  |^|_ x  e.  B  ( F `  x )  =  |^|_ x  e.  B  ( ( F  |`  B ) `  x )
7 df-ima 4868 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
87inteqi 4147 . 2  |-  |^| ( F " B )  = 
|^| ran  ( F  |`  B )
93, 6, 83eqtr4g 2500 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 1369    C_ wss 3343   |^|cint 4143   |^|_ciin 4187   ran crn 4856    |` cres 4857   "cima 4858    Fn wfn 5428   ` cfv 5433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-int 4144  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-fv 5441
This theorem is referenced by:  elrfirn  29050
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