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Theorem resima2 5881
Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
Assertion
Ref Expression
resima2 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Proof of Theorem resima2
StepHypRef Expression
1 sseqin2 4189 . . . 4 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
2 reseq2 5841 . . . 4 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
31, 2sylbi 218 . . 3 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
43rneqd 5801 . 2 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴𝐵))
5 df-ima 5561 . . 3 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
6 resres 5859 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
76rneqi 5800 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
85, 7eqtri 2841 . 2 ((𝐴𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
9 df-ima 5561 . 2 (𝐴𝐵) = ran (𝐴𝐵)
104, 8, 93eqtr4g 2878 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cin 3932  wss 3933  ran crn 5549  cres 5550  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  ressuppss  7838  ressuppssdif  7840  marypha1lem  8885  ackbij2lem3  9651  dmdprdsplit2lem  19096  cnpresti  21824  cnprest  21825  limcflf  24406  limcresi  24410  limciun  24419  efopnlem2  25167  pthhashvtx  32271  cvmopnlem  32422  cvmlift2lem9a  32447  poimirlem4  34777  limsupresre  41853  limsupresico  41857  liminfresico  41928
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