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Theorem resima2 4671
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 4385 . 2 ((𝐴𝐶) “ 𝐵) = ran ((𝐴𝐶) ↾ 𝐵)
2 resres 4651 . . . 4 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
32rneqi 4589 . . 3 ran ((𝐴𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶𝐵))
4 df-ss 2958 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
5 incom 3156 . . . . . . . 8 (𝐶𝐵) = (𝐵𝐶)
65a1i 9 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐶𝐵) = (𝐵𝐶))
76reseq2d 4639 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴 ↾ (𝐵𝐶)))
87rneqd 4590 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = ran (𝐴 ↾ (𝐵𝐶)))
9 reseq2 4634 . . . . . . 7 ((𝐵𝐶) = 𝐵 → (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
109rneqd 4590 . . . . . 6 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = ran (𝐴𝐵))
11 df-ima 4385 . . . . . 6 (𝐴𝐵) = ran (𝐴𝐵)
1210, 11syl6eqr 2106 . . . . 5 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐵𝐶)) = (𝐴𝐵))
138, 12eqtrd 2088 . . . 4 ((𝐵𝐶) = 𝐵 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
144, 13sylbi 118 . . 3 (𝐵𝐶 → ran (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
153, 14syl5eq 2100 . 2 (𝐵𝐶 → ran ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
161, 15syl5eq 2100 1 (𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cin 2943  wss 2944  ran crn 4373  cres 4374  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by: (None)
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