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Theorem elfunsg 34883
Description: Closed form of elfuns 34882. (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 2821 . 2 (𝑓 = 𝐹 → (𝑓 Funs 𝐹 Funs ))
2 funeq 6568 . 2 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
3 vex 3478 . . 3 𝑓 ∈ V
43elfuns 34882 . 2 (𝑓 Funs ↔ Fun 𝑓)
51, 2, 4vtoclbg 3559 1 (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  Fun wfun 6537   Funs cfuns 34804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7974  df-2nd 7975  df-txp 34821  df-fix 34826  df-funs 34828
This theorem is referenced by: (None)
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