Theorem imakeq1 4225

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

Assertion

Ref	Expression
imakeq1	⊢ (A = B → (A “k C) = (B “k C))

Proof of Theorem imakeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 ⊢ (A = B → (⟪y, x⟫ ∈ A ↔ ⟪y, x⟫ ∈ B))
2	1	rexbidv 2636	. . 3 ⊢ (A = B → (∃y ∈ C ⟪y, x⟫ ∈ A ↔ ∃y ∈ C ⟪y, x⟫ ∈ B))
3	2	abbidv 2468	. 2 ⊢ (A = B → {x ∣ ∃y ∈ C ⟪y, x⟫ ∈ A} = {x ∣ ∃y ∈ C ⟪y, x⟫ ∈ B})
4		df-imak 4190	. 2 ⊢ (A “k C) = {x ∣ ∃y ∈ C ⟪y, x⟫ ∈ A}
5		df-imak 4190	. 2 ⊢ (B “k C) = {x ∣ ∃y ∈ C ⟪y, x⟫ ∈ B}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → (A “k C) = (B “k C))

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ⟪copk 4058   “_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:  imakeq1i  4227  imakeq1d  4229