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Theorem inpreima 6832
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 6420 . 2 (Fun 𝐹 → Fun 𝐹)
2 imain 6438 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
31, 2syl 17 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  cin 3939  ccnv 5553  cima 5557  Fun wfun 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6356
This theorem is referenced by:  fimacnvinrn  6838  frnsuppeq  7838  ofco2  20995  cnrest2  21829  cnhaus  21897  kgencn3  22101  qtoptop2  22242  basqtop  22254  ismbfd  24174  mbfimaopn2  24192  i1fima  24213  i1fima2  24214  i1fd  24216  disjpreima  30268  sspreima  30326  smfco  42962
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