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Theorem elfunsg 32992
 Description: Closed form of elfuns 32991. (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 2870 . 2 (𝑓 = 𝐹 → (𝑓 Funs 𝐹 Funs ))
2 funeq 6250 . 2 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
3 vex 3440 . . 3 𝑓 ∈ V
43elfuns 32991 . 2 (𝑓 Funs ↔ Fun 𝑓)
51, 2, 4vtoclbg 3511 1 (𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2081  Fun wfun 6224   Funs cfuns 32913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-eprel 5358  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-fo 6236  df-fv 6238  df-1st 7550  df-2nd 7551  df-txp 32930  df-fix 32935  df-funs 32937 This theorem is referenced by: (None)
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