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Theorem resima2 4860
 Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
resima2 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Proof of Theorem resima2
StepHypRef Expression
1 df-ima 4559 . 2 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
2 resres 4838 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
32rneqi 4774 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
4 df-ss 3088 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
5 incom 3272 . . . . . . . 8 (𝐶𝐵) = (𝐵𝐶)
65a1i 9 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐶𝐵) = (𝐵𝐶))
76reseq2d 4826 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴 ↾ (𝐵𝐶)))
87rneqd 4775 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴 ↾ (𝐵𝐶)))
9 reseq2 4821 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
109rneqd 4775 . . . . . 6 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = ran (𝐴𝐵))
11 df-ima 4559 . . . . . 6 (𝐴𝐵) = ran (𝐴𝐵)
1210, 11eqtr4di 2191 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
138, 12eqtrd 2173 . . . 4 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
144, 13sylbi 120 . . 3 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
153, 14syl5eq 2185 . 2 (𝐵𝐶 → ran ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
161, 15syl5eq 2185 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∩ cin 3074   ⊆ wss 3075  ran crn 4547   ↾ cres 4548   “ cima 4549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559 This theorem is referenced by:  cnptopresti  12444  cnptoprest  12445
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