Theorem elfv 6662

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 6659	. . 3 ⊢ (𝐹‘𝐵) = ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)}
2	1	eleq2i 2904	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ 𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)})
3		eluniab 4842	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
4	2, 3	bitri 277	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776   ∈ wcel 2110  {cab 2799  ∪ cuni 4831   class class class wbr 5058  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-sn 4561  df-uni 4832  df-iota 6308  df-fv 6357

This theorem is referenced by:  fv3  6682