Theorem cokeq2 4231

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

Assertion

Ref	Expression
cokeq2	⊢ (A = B → (C ∘k A) = (C ∘k B))

Proof of Theorem cokeq2

Step	Hyp	Ref	Expression
1		cnvkeq 4215	. . . . 5 ⊢ (A = B → ◡kA = ◡kB)
2	1	ins3keqd 4223	. . . 4 ⊢ (A = B → Ins3k ◡kA = Ins3k ◡kB)
3	2	ineq2d 3457	. . 3 ⊢ (A = B → ( Ins2k C ∩ Ins3k ◡kA) = ( Ins2k C ∩ Ins3k ◡kB))
4	3	imakeq1d 4228	. 2 ⊢ (A = B → (( Ins2k C ∩ Ins3k ◡kA) “k V) = (( Ins2k C ∩ Ins3k ◡kB) “k V))
5		df-cok 4190	. 2 ⊢ (C ∘k A) = (( Ins2k C ∩ Ins3k ◡kA) “k V)
6		df-cok 4190	. 2 ⊢ (C ∘k B) = (( Ins2k C ∩ Ins3k ◡kB) “k V)
7	4, 5, 6	3eqtr4g 2410	1 ⊢ (A = B → (C ∘k A) = (C ∘k B))

Syntax hints:   → wi 4   = wceq 1642  Vcvv 2859   ∩ cin 3208  ^◡_kccnvk 4175   Ins2_k cins2k 4176   Ins3_k cins3k 4177   “_k cimak 4179   ∘_k ccomk 4180

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 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-cnvk 4186  df-ins3k 4188  df-imak 4189  df-cok 4190

This theorem is referenced by:  cokeq2i  4233  cokeq2d  4235