Theorem elfuns 5830

Description: Membership in the set of all functions. (Contributed by SF, 23-Feb-2015.)

Hypothesis

Ref	Expression
elfuns.1	⊢ F ∈ V

Assertion

Ref	Expression
elfuns	⊢ (F ∈ Funs ↔ Fun F)

Proof of Theorem elfuns

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfuns.1	. 2 ⊢ F ∈ V
2		funeq 5128	. 2 ⊢ (f = F → (Fun f ↔ Fun F))
3		df-funs 5761	. 2 ⊢ Funs = {f ∣ Fun f}
4	1, 2, 3	elab2 2989	1 ⊢ (F ∈ Funs ↔ Fun F)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2860  Fun wfun 4776   Funs cfuns 5760

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-nan 1288  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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-fun 4790  df-funs 5761

This theorem is referenced by:  elfunsg  5831  fnsex  5833  mapexi  6004  fnpm  6009