Theorem elfv 6877

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 6874	. . 3 ⊢ (𝐹‘𝐵) = ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)}
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ 𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)})
3		eluniab 4881	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∪ cuni 4867   class class class wbr 5103  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3916  df-sn 4585  df-uni 4868  df-iota 6489  df-fv 6541

This theorem is used by:  fv3  6897