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Theorem resima2 4587
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 4301 . 2 ((A𝐶) “ B) = ran ((A𝐶) ↾ B)
2 resres 4567 . . . 4 ((A𝐶) ↾ B) = (A ↾ (𝐶B))
32rneqi 4505 . . 3 ran ((A𝐶) ↾ B) = ran (A ↾ (𝐶B))
4 df-ss 2925 . . . 4 (B𝐶 ↔ (B𝐶) = B)
5 incom 3123 . . . . . . . 8 (𝐶B) = (B𝐶)
65a1i 9 . . . . . . 7 ((B𝐶) = B → (𝐶B) = (B𝐶))
76reseq2d 4555 . . . . . 6 ((B𝐶) = B → (A ↾ (𝐶B)) = (A ↾ (B𝐶)))
87rneqd 4506 . . . . 5 ((B𝐶) = B → ran (A ↾ (𝐶B)) = ran (A ↾ (B𝐶)))
9 reseq2 4550 . . . . . . 7 ((B𝐶) = B → (A ↾ (B𝐶)) = (AB))
109rneqd 4506 . . . . . 6 ((B𝐶) = B → ran (A ↾ (B𝐶)) = ran (AB))
11 df-ima 4301 . . . . . 6 (AB) = ran (AB)
1210, 11syl6eqr 2087 . . . . 5 ((B𝐶) = B → ran (A ↾ (B𝐶)) = (AB))
138, 12eqtrd 2069 . . . 4 ((B𝐶) = B → ran (A ↾ (𝐶B)) = (AB))
144, 13sylbi 114 . . 3 (B𝐶 → ran (A ↾ (𝐶B)) = (AB))
153, 14syl5eq 2081 . 2 (B𝐶 → ran ((A𝐶) ↾ B) = (AB))
161, 15syl5eq 2081 1 (B𝐶 → ((A𝐶) “ B) = (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cin 2910  wss 2911  ran crn 4289  cres 4290  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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