Theorem dfima4 4953

Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 14-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
dfima4	⊢ (A “ B) = {y ∣ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A)}

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem dfima4

Step	Hyp	Ref	Expression
1		df-ima 4728	. 2 ⊢ (A “ B) = {y ∣ ∃x ∈ B xAy}
2		df-br 4641	. . . . 5 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
3	2	rexbii 2640	. . . 4 ⊢ (∃x ∈ B xAy ↔ ∃x ∈ B ⟨x, y⟩ ∈ A)
4		df-rex 2621	. . . 4 ⊢ (∃x ∈ B ⟨x, y⟩ ∈ A ↔ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A))
5	3, 4	bitri 240	. . 3 ⊢ (∃x ∈ B xAy ↔ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A))
6	5	abbii 2466	. 2 ⊢ {y ∣ ∃x ∈ B xAy} = {y ∣ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A)}
7	1, 6	eqtri 2373	1 ⊢ (A “ B) = {y ∣ ∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A)}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ⟨cop 4562   class class class wbr 4640   “ cima 4723

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rex 2621  df-br 4641  df-ima 4728

This theorem is referenced by:  imassrn  5010  imai  5011