Theorem cokeq2 4232

Description: Equality theorem for Kuratowski composition of two classes. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
cokeq2	|- (A = B -> (C o.k A) = (C o.k B))

Proof of Theorem cokeq2

Step	Hyp	Ref	Expression
1		cnvkeq 4216	. . . . 5 |- (A = B -> `'kA = `'kB)
2	1	ins3keqd 4224	. . . 4 |- (A = B -> Ins3k `'kA = Ins3k `'kB)
3	2	ineq2d 3458	. . 3 |- (A = B -> ( Ins2k C i^i Ins3k `'kA) = ( Ins2k C i^i Ins3k `'kB))
4	3	imakeq1d 4229	. 2 |- (A = B -> (( Ins2k C i^i Ins3k `'kA)"k_V) = (( Ins2k C i^i Ins3k `'kB)"k_V))
5		df-cok 4191	. 2 |- (C o.k A) = (( Ins2k C i^i Ins3k `'kA)"k_V)
6		df-cok 4191	. 2 |- (C o.k B) = (( Ins2k C i^i Ins3k `'kB)"k_V)
7	4, 5, 6	3eqtr4g 2410	1 |- (A = B -> (C o.k A) = (C o.k B))

Syntax hints:   -> wi 4   = wceq 1642  _Vcvv 2860   i^i cin 3209  `'_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   o._k ccomk 4181

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-cnvk 4187  df-ins3k 4189  df-imak 4190  df-cok 4191

This theorem is referenced by:  cokeq2i  4234  cokeq2d  4236