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Theorem resima2 4733
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 4441 . 2 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
2 resres 4713 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
32rneqi 4651 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
4 df-ss 3010 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
5 incom 3190 . . . . . . . 8 (𝐶𝐵) = (𝐵𝐶)
65a1i 9 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐶𝐵) = (𝐵𝐶))
76reseq2d 4701 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴 ↾ (𝐵𝐶)))
87rneqd 4652 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴 ↾ (𝐵𝐶)))
9 reseq2 4696 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
109rneqd 4652 . . . . . 6 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = ran (𝐴𝐵))
11 df-ima 4441 . . . . . 6 (𝐴𝐵) = ran (𝐴𝐵)
1210, 11syl6eqr 2138 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
138, 12eqtrd 2120 . . . 4 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
144, 13sylbi 119 . . 3 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
153, 14syl5eq 2132 . 2 (𝐵𝐶 → ran ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
161, 15syl5eq 2132 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cin 2996  wss 2997  ran crn 4429  cres 4430  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by: (None)
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