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Theorem inpreima 5948
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 5583 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imain 5601 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
31, 2syl 17 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 1405    i^i cin 3412   `'ccnv 4941   "cima 4945   Fun wfun 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-fun 5527
This theorem is referenced by:  frnsuppeq  6868  nn0suppOLD  10811  ofco2  19137  cnrest2  19972  cnhaus  20040  kgencn3  20243  qtoptop2  20384  basqtop  20396  ismbfd  22231  mbfimaopn2  22248  i1fima  22269  i1fima2  22270  i1fd  22272  disjpreima  27756  fimacnvinrn  27798  sspreima  27808  fsuppeq  35386
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