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Theorem funimacnv 5167
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 5166 . . . 4 (Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
21rneqd 4736 . . 3 (Fun 𝐹 → ran (𝐹𝐴) = ran (𝐹 ↾ (𝐹𝐴)))
3 df-ima 4520 . . 3 (𝐹 “ (𝐹𝐴)) = ran (𝐹 ↾ (𝐹𝐴))
42, 3syl6reqr 2167 . 2 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = ran (𝐹𝐴))
5 df-rn 4518 . . . 4 ran 𝐹 = dom 𝐹
65ineq2i 3242 . . 3 (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom 𝐹)
7 dmres 4808 . . 3 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 dfdm4 4699 . . 3 dom (𝐹𝐴) = ran (𝐹𝐴)
96, 7, 83eqtr2ri 2143 . 2 ran (𝐹𝐴) = (𝐴 ∩ ran 𝐹)
104, 9syl6eq 2164 1 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cin 3038  ccnv 4506  dom cdm 4507  ran crn 4508  cres 4509  cima 4510  Fun wfun 5085
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093
This theorem is referenced by:  funimass1  5168  funimass2  5169
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