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Theorem resima2 5891
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 4135 . . . 4 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
2 reseq2 5851 . . . 4 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
31, 2sylbi 220 . . 3 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
43rneqd 5812 . 2 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴𝐵))
5 df-ima 5569 . . 3 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
6 resres 5869 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
76rneqi 5811 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
85, 7eqtri 2765 . 2 ((𝐴𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
9 df-ima 5569 . 2 (𝐴𝐵) = ran (𝐴𝐵)
104, 8, 93eqtr4g 2803 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  cin 3870  wss 3871  ran crn 5557  cres 5558  cima 5559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-br 5059  df-opab 5121  df-xp 5562  df-rel 5563  df-cnv 5564  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569
This theorem is referenced by:  ressuppss  7930  ressuppssdif  7932  marypha1lem  9054  ackbij2lem3  9860  dmdprdsplit2lem  19437  cnpresti  22190  cnprest  22191  limcflf  24783  limcresi  24787  limciun  24796  efopnlem2  25550  pthhashvtx  32807  cvmopnlem  32958  cvmlift2lem9a  32983  naddcllem  33573  negsval  33865  poimirlem4  35523  limsupresre  42920  limsupresico  42924  liminfresico  42995
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