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Theorem elfunsg 35897
Description: Closed form of elfuns 35896. (Contributed by Scott Fenton, 2-May-2014.)
Assertion
Ref Expression
elfunsg (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))

Proof of Theorem elfunsg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2826 . 2 (𝑓 = 𝐹 → (𝑓 Funs 𝐹 Funs ))
2 funeq 6587 . 2 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
3 vex 3481 . . 3 𝑓 ∈ V
43elfuns 35896 . 2 (𝑓 Funs ↔ Fun 𝑓)
51, 2, 4vtoclbg 3556 1 (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2105  Fun wfun 6556   Funs cfuns 35818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-fix 35840  df-funs 35842
This theorem is referenced by: (None)
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