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Theorem inpreima 5280
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 4945 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imain 4968 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
31, 2syl 14 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 1243    i^i cin 2913   `'ccnv 4331   "cima 4335   Fun wfun 4883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-fun 4891
This theorem is referenced by:  nn0supp  8206
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