Theorem imagekeq 4245

Description: Equality theorem for image operation. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
imagekeq	|- (A = B -> ImagekA = ImagekB)

Proof of Theorem imagekeq

Step	Hyp	Ref	Expression
1		sikeq 4242	. . . . . . . 8 |- (A = B -> SIk A = SIk B)
2	1	cnvkeqd 4218	. . . . . . 7 |- (A = B -> `'k SIk A = `'k SIk B)
3	2	cokeq2d 4236	. . . . . 6 |- (A = B -> ( Sk o.k `'k SIk A) = ( Sk o.k `'k SIk B))
4	3	ins3keqd 4224	. . . . 5 |- (A = B -> Ins3k ( Sk o.k `'k SIk A) = Ins3k ( Sk o.k `'k SIk B))
5	4	symdifeq2d 3256	. . . 4 |- (A = B -> ( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk A)) = ( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk B)))
6	5	imakeq1d 4229	. . 3 |- (A = B -> (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk A))"k~P1 ~P1 1c) = (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk B))"k~P1 ~P1 1c))
7	6	difeq2d 3386	. 2 |- (A = B -> ((_V X.k _V) \ (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk A))"k~P1 ~P1 1c)) = ((_V X.k _V) \ (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk B))"k~P1 ~P1 1c)))
8		df-imagek 4195	. 2 |- ImagekA = ((_V X.k _V) \ (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk A))"k~P1 ~P1 1c))
9		df-imagek 4195	. 2 |- ImagekB = ((_V X.k _V) \ (( Ins2k Sk (+) Ins3k ( Sk o.k `'k SIk B))"k~P1 ~P1 1c))
10	7, 8, 9	3eqtr4g 2410	1 |- (A = B -> ImagekA = ImagekB)

Syntax hints:   -> wi 4   = wceq 1642  _Vcvv 2860   \ cdif 3207   (+) csymdif 3210  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175  `'_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   o._k ccomk 4181   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-cnvk 4187  df-ins3k 4189  df-imak 4190  df-cok 4191  df-sik 4193  df-imagek 4195

This theorem is referenced by: (None)