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Theorem inpreima 5668
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 5324 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imain 5344 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
31, 2syl 15 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 1632    i^i cin 3164   `'ccnv 4704   "cima 4708   Fun wfun 5265
This theorem is referenced by:  nn0supp  10033  cnrest2  17030  cnhaus  17098  kgencn3  17269  qtoptop2  17406  basqtop  17418  ismbfd  19011  mbfimaopn2  19028  i1fima  19049  i1fima2  19050  i1fd  19052  fimacnvinrn  23214  sspreima  23225  disjpreima  23376  inpreimaOLD  26470  fsuppeq  27362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
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