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Theorem difpreima 5943
Description: Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
Assertion
Ref Expression
difpreima  |-  ( Fun 
F  ->  ( `' F " ( A  \  B ) )  =  ( ( `' F " A )  \  ( `' F " B ) ) )

Proof of Theorem difpreima
StepHypRef Expression
1 funcnvcnv 5587 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imadif 5604 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A 
\  B ) )  =  ( ( `' F " A ) 
\  ( `' F " B ) ) )
31, 2syl 16 1  |-  ( Fun 
F  ->  ( `' F " ( A  \  B ) )  =  ( ( `' F " A )  \  ( `' F " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    \ cdif 3436   `'ccnv 4950   "cima 4954   Fun wfun 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-fun 5531
This theorem is referenced by:  gsumpropd2lem  15627  fsumcvg4  26545  zrhunitpreima  26572  imambfm  26841  sibfof  26890  eulerpartlemmf  26922  dvtanlem  28609  itg2addnclem  28611  itg2addnclem2  28612
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