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Theorem funimacnv 4918
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 4917 . . . 4 (Fun 𝐹(𝐹A) = (𝐹 ↾ (𝐹A)))
21rneqd 4506 . . 3 (Fun 𝐹 → ran (𝐹A) = ran (𝐹 ↾ (𝐹A)))
3 df-ima 4301 . . 3 (𝐹 “ (𝐹A)) = ran (𝐹 ↾ (𝐹A))
42, 3syl6reqr 2088 . 2 (Fun 𝐹 → (𝐹 “ (𝐹A)) = ran (𝐹A))
5 df-rn 4299 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 3129 . . 3 (A ∩ ran 𝐹) = (A ∩ dom 𝐹)
7 dmres 4575 . . 3 dom (𝐹A) = (A ∩ dom 𝐹)
8 dfdm4 4470 . . 3 dom (𝐹A) = ran (𝐹A)
96, 7, 83eqtr2ri 2064 . 2 ran (𝐹A) = (A ∩ ran 𝐹)
104, 9syl6eq 2085 1 (Fun 𝐹 → (𝐹 “ (𝐹A)) = (A ∩ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cin 2910  ccnv 4287  dom cdm 4288  ran crn 4289  cres 4290  cima 4291  Fun wfun 4839
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  funimass1  4919  funimass2  4920
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