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Theorem elfunsg 35545
Description: Closed form of elfuns 35544. (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 2817 . 2 (𝑓 = 𝐹 → (𝑓 Funs 𝐹 Funs ))
2 funeq 6578 . 2 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
3 vex 3477 . . 3 𝑓 ∈ V
43elfuns 35544 . 2 (𝑓 Funs ↔ Fun 𝑓)
51, 2, 4vtoclbg 3544 1 (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  Fun wfun 6547   Funs cfuns 35466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-1st 7999  df-2nd 8000  df-txp 35483  df-fix 35488  df-funs 35490
This theorem is referenced by: (None)
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