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Theorem inpreima 5999
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 5637 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imain 5655 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1374    i^i cin 3468   `'ccnv 4991   "cima 4995   Fun wfun 5573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581
This theorem is referenced by:  frnsuppeq  6903  nn0suppOLD  10839  ofco2  18713  cnrest2  19546  cnhaus  19614  kgencn3  19787  qtoptop2  19928  basqtop  19940  ismbfd  21775  mbfimaopn2  21792  i1fima  21813  i1fima2  21814  i1fd  21816  disjpreima  27104  fimacnvinrn  27134  sspreima  27143  fsuppeq  30636
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