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Theorem resima2 6017
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 4216 . . . 4 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
2 reseq2 5977 . . . 4 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
31, 2sylbi 216 . . 3 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
43rneqd 5938 . 2 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴𝐵))
5 df-ima 5690 . . 3 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
6 resres 5995 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
76rneqi 5937 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
85, 7eqtri 2761 . 2 ((𝐴𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
9 df-ima 5690 . 2 (𝐴𝐵) = ran (𝐴𝐵)
104, 8, 93eqtr4g 2798 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3948  wss 3949  ran crn 5678  cres 5679  cima 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  ressuppss  8168  ressuppssdif  8170  naddcllem  8675  marypha1lem  9428  ackbij2lem3  10236  eqg0subgecsn  19074  dmdprdsplit2lem  19915  cnpresti  22792  cnprest  22793  limcflf  25398  limcresi  25402  limciun  25411  efopnlem2  26165  negsval  27503  pthhashvtx  34149  cvmopnlem  34300  cvmlift2lem9a  34325  poimirlem4  36540  limsupresre  44460  limsupresico  44464  liminfresico  44535
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