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Theorem imagekeq 4244
 Description: Equality theorem for image operation. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
imagekeq (A = B → ImagekA = ImagekB)

Proof of Theorem imagekeq
StepHypRef Expression
1 sikeq 4241 . . . . . . . 8 (A = BSIk A = SIk B)
21cnvkeqd 4217 . . . . . . 7 (A = Bk SIk A = k SIk B)
32cokeq2d 4235 . . . . . 6 (A = B → ( Sk k k SIk A) = ( Sk k k SIk B))
43ins3keqd 4223 . . . . 5 (A = BIns3k ( Sk k k SIk A) = Ins3k ( Sk k k SIk B))
54symdifeq2d 3255 . . . 4 (A = B → ( Ins2k SkIns3k ( Sk k k SIk A)) = ( Ins2k SkIns3k ( Sk k k SIk B)))
65imakeq1d 4228 . . 3 (A = B → (( Ins2k SkIns3k ( Sk k k SIk A)) “k 111c) = (( Ins2k SkIns3k ( Sk k k SIk B)) “k 111c))
76difeq2d 3385 . 2 (A = B → ((V ×k V) (( Ins2k SkIns3k ( Sk k k SIk A)) “k 111c)) = ((V ×k V) (( Ins2k SkIns3k ( Sk k k SIk B)) “k 111c)))
8 df-imagek 4194 . 2 ImagekA = ((V ×k V) (( Ins2k SkIns3k ( Sk k k SIk A)) “k 111c))
9 df-imagek 4194 . 2 ImagekB = ((V ×k V) (( Ins2k SkIns3k ( Sk k k SIk B)) “k 111c))
107, 8, 93eqtr4g 2410 1 (A = B → ImagekA = ImagekB)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ⊕ csymdif 3209  1cc1c 4134  ℘1cpw1 4135   ×k cxpk 4174  ◡kccnvk 4175   Ins2k cins2k 4176   Ins3k cins3k 4177   “k cimak 4179   ∘k ccomk 4180   SIk csik 4181  Imagekcimagek 4182   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-cnvk 4186  df-ins3k 4188  df-imak 4189  df-cok 4190  df-sik 4192  df-imagek 4194 This theorem is referenced by: (None)
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