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Theorem resima2 5352
 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 3779 . . . 4 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
2 reseq2 5312 . . . 4 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
31, 2sylbi 206 . . 3 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
43rneqd 5274 . 2 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴𝐵))
5 df-ima 5051 . . 3 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
6 resres 5329 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
76rneqi 5273 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
85, 7eqtri 2632 . 2 ((𝐴𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
9 df-ima 5051 . 2 (𝐴𝐵) = ran (𝐴𝐵)
104, 8, 93eqtr4g 2669 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539   ⊆ wss 3540  ran crn 5039   ↾ cres 5040   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  ressuppss  7201  ressuppssdif  7203  marypha1lem  8222  ackbij2lem3  8946  dmdprdsplit2lem  18267  cnpresti  20902  cnprest  20903  limcflf  23451  limcresi  23455  limciun  23464  efopnlem2  24203  cvmopnlem  30514  cvmlift2lem9a  30539  poimirlem4  32583
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