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Theorem funimacnv 4975
 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 funcnvres2 4974 . . . 4 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
21rneqd 4563 . . 3 (Fun 𝐹 → ran (𝐹𝐴) = ran (𝐹 ↾ (𝐹𝐴)))
3 df-ima 4358 . . 3 (𝐹 “ (𝐹𝐴)) = ran (𝐹 ↾ (𝐹𝐴))
42, 3syl6reqr 2091 . 2 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = ran (𝐹𝐴))
5 df-rn 4356 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 3135 . . 3 (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom 𝐹)
7 dmres 4632 . . 3 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 dfdm4 4527 . . 3 dom (𝐹𝐴) = ran (𝐹𝐴)
96, 7, 83eqtr2ri 2067 . 2 ran (𝐹𝐴) = (𝐴 ∩ ran 𝐹)
104, 9syl6eq 2088 1 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∩ cin 2916  ◡ccnv 4344  dom cdm 4345  ran crn 4346   ↾ cres 4347   “ cima 4348  Fun wfun 4896 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 3875  ax-pow 3927  ax-pr 3944 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 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904 This theorem is referenced by:  funimass1  4976  funimass2  4977
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