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Theorem elfunsg 5830
 Description: Membership in the set of all functions. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
elfunsg (F V → (F Funs ↔ Fun F))

Proof of Theorem elfunsg
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . 2 (f = F → (f FunsF Funs ))
2 funeq 5127 . 2 (f = F → (Fun f ↔ Fun F))
3 vex 2862 . . 3 f V
43elfuns 5829 . 2 (f Funs ↔ Fun f)
51, 2, 4vtoclbg 2915 1 (F V → (F Funs ↔ Fun F))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  Fun wfun 4775   Funs cfuns 5759 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-fun 4789  df-funs 5760 This theorem is referenced by:  elfunsi  5831
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