Theorem elfv 5327

Description: Membership in a function value. (Contributed by set.mm contributors, 30-Apr-2004.)

Assertion

Ref	Expression
elfv	⊢ (A ∈ (F ‘B) ↔ ∃x(A ∈ x ∧ ∀y(BFy ↔ y = x)))

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

Allowed substitution hint:   A(y)

Proof of Theorem elfv

Step	Hyp	Ref	Expression
1		fv2 5325	. . 3 ⊢ (F ‘B) = ∪{x ∣ ∀y(BFy ↔ y = x)}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ (F ‘B) ↔ A ∈ ∪{x ∣ ∀y(BFy ↔ y = x)})
3		eluniab 3904	. 2 ⊢ (A ∈ ∪{x ∣ ∀y(BFy ↔ y = x)} ↔ ∃x(A ∈ x ∧ ∀y(BFy ↔ y = x)))
4	2, 3	bitri 240	1 ⊢ (A ∈ (F ‘B) ↔ ∃x(A ∈ x ∧ ∀y(BFy ↔ y = x)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  {cab 2339  ∪cuni 3892   class class class wbr 4640   ‘cfv 4782

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 2479  df-rex 2621  df-v 2862  df-sn 3742  df-uni 3893  df-iota 4340  df-fv 4796

This theorem is referenced by:  fv3  5342