Theorem imakeq1d 4229

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

Hypothesis

Ref	Expression
imakeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
imakeq1d	⊢ (φ → (A “k C) = (B “k C))

Proof of Theorem imakeq1d

Step	Hyp	Ref	Expression
1		imakeq1d.1	. 2 ⊢ (φ → A = B)
2		imakeq1 4225	. 2 ⊢ (A = B → (A “k C) = (B “k C))
3	1, 2	syl 15	1 ⊢ (φ → (A “k C) = (B “k C))

This proof depends on syntax axioms:   → wi 4   = wceq 1642   “_k cimak 4180

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2621  df-imak 4190

This theorem is used by:  cokeq1  4231  cokeq2  4232  imagekeq  4245  addceq2  4385