Theorem elfv 6643

Description: Membership in a function value. (Contributed by NM, 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 6640	. . 3 ⊢ (𝐹‘𝐵) = ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)}
2	1	eleq2i 2881	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ 𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)})
3		eluniab 4815	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
4	2, 3	bitri 278	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))

Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∪ cuni 4800   class class class wbr 5030  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-uni 4801  df-iota 6283  df-fv 6332

This theorem is referenced by:  fv3  6663