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Theorem inpreima 5239
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 “ (AB)) = ((𝐹A) ∩ (𝐹B)))

Proof of Theorem inpreima
StepHypRef Expression
1 funcnvcnv 4904 . 2 (Fun 𝐹 → Fun 𝐹)
2 imain 4927 . 2 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∩ (𝐹B)))
31, 2syl 14 1 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∩ (𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  cin 2913  ccnv 4290  cima 4294  Fun wfun 4842
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 3869  ax-pow 3921  ax-pr 3938
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 3356  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-fun 4850
This theorem is referenced by:  nn0supp  8102
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