Theorem elfv 5556

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 5553	. . 3 ⊢ (𝐹‘𝐵) = ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)}
2	1	eleq2i 2263	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ 𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)})
3		eluniab 3851	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
4	2, 3	bitri 184	1 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1362  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∪ cuni 3839   class class class wbr 4033  ‘cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sn 3628  df-uni 3840  df-iota 5219  df-fv 5266

This theorem is referenced by:  fv3  5581  relelfvdm  5590