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Theorem funimacnv 5247
Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimacnv (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))

Proof of Theorem funimacnv
StepHypRef Expression
1 df-ima 4600 . . 3 (𝐹 “ (𝐹𝐴)) = ran (𝐹 ↾ (𝐹𝐴))
2 funcnvres2 5246 . . . 4 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
32rneqd 4816 . . 3 (Fun 𝐹 → ran (𝐹𝐴) = ran (𝐹 ↾ (𝐹𝐴)))
41, 3eqtr4id 2209 . 2 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = ran (𝐹𝐴))
5 df-rn 4598 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 3305 . . 3 (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom 𝐹)
7 dmres 4888 . . 3 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 dfdm4 4779 . . 3 dom (𝐹𝐴) = ran (𝐹𝐴)
96, 7, 83eqtr2ri 2185 . 2 ran (𝐹𝐴) = (𝐴 ∩ ran 𝐹)
104, 9eqtrdi 2206 1 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  cin 3101  ccnv 4586  dom cdm 4587  ran crn 4588  cres 4589  cima 4590  Fun wfun 5165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-fun 5173
This theorem is referenced by:  funimass1  5248  funimass2  5249
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