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Theorem elfunsg 35917
Description: Closed form of elfuns 35916. (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 2829 . 2 (𝑓 = 𝐹 → (𝑓 Funs 𝐹 Funs ))
2 funeq 6586 . 2 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
3 vex 3484 . . 3 𝑓 ∈ V
43elfuns 35916 . 2 (𝑓 Funs ↔ Fun 𝑓)
51, 2, 4vtoclbg 3557 1 (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  Fun wfun 6555   Funs cfuns 35838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-fix 35860  df-funs 35862
This theorem is referenced by: (None)
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