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Theorem inpreima 5859
Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
Assertion
Ref Expression
inpreima  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )

Proof of Theorem inpreima
StepHypRef Expression
1 funcnvcnv 5511 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imain 5531 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
31, 2syl 16 1  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    i^i cin 3321   `'ccnv 4879   "cima 4883   Fun wfun 5450
This theorem is referenced by:  nn0supp  10275  cnrest2  17352  cnhaus  17420  kgencn3  17592  qtoptop2  17733  basqtop  17745  ismbfd  19534  mbfimaopn2  19551  i1fima  19572  i1fima2  19573  i1fd  19575  disjpreima  24028  fimacnvinrn  24049  sspreima  24059  fsuppeq  27238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458
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