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Theorem resima2 5430
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 3815 . . . 4 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
2 reseq2 5389 . . . 4 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
31, 2sylbi 207 . . 3 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
43rneqd 5351 . 2 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴𝐵))
5 df-ima 5125 . . 3 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
6 resres 5407 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
76rneqi 5350 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
85, 7eqtri 2643 . 2 ((𝐴𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
9 df-ima 5125 . 2 (𝐴𝐵) = ran (𝐴𝐵)
104, 8, 93eqtr4g 2680 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  cin 3571  wss 3572  ran crn 5113  cres 5114  cima 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by:  ressuppss  7311  ressuppssdif  7313  marypha1lem  8336  ackbij2lem3  9060  dmdprdsplit2lem  18438  cnpresti  21086  cnprest  21087  limcflf  23639  limcresi  23643  limciun  23652  efopnlem2  24397  cvmopnlem  31245  cvmlift2lem9a  31270  poimirlem4  33393  limsupresre  39734  limsupresico  39738  liminfresico  39803
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