Theorem imaeq2 4938

Description: Equality theorem for image. (Contributed by set.mm contributors, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	⊢ (A = B → (C “ A) = (C “ B))

Proof of Theorem imaeq2

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

Step	Hyp	Ref	Expression
1		rexeq 2808	. . 3 ⊢ (A = B → (∃y ∈ A yCx ↔ ∃y ∈ B yCx))
2	1	abbidv 2467	. 2 ⊢ (A = B → {x ∣ ∃y ∈ A yCx} = {x ∣ ∃y ∈ B yCx})
3		df-ima 4727	. 2 ⊢ (C “ A) = {x ∣ ∃y ∈ A yCx}
4		df-ima 4727	. 2 ⊢ (C “ B) = {x ∣ ∃y ∈ B yCx}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → (C “ A) = (C “ B))

Syntax hints:   → wi 4   = wceq 1642  {cab 2339  ∃wrex 2615   class class class wbr 4639   “ cima 4722

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-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-ima 4727

This theorem is referenced by:  imaeq2i  4940  imaeq2d  4942  foima  5274  f1imacnv  5302  clos1induct  5880  ecexr  5950  ncdisjun  6136