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Theorem fullfunfnv 33400
Description: The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fullfunfnv FullFun𝐹 Fn V

Proof of Theorem fullfunfnv
StepHypRef Expression
1 funpartfun 33397 . . . . 5 Fun Funpart𝐹
2 funfn 6378 . . . . 5 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
31, 2mpbi 232 . . . 4 Funpart𝐹 Fn dom Funpart𝐹
4 0ex 5202 . . . . . 6 ∅ ∈ V
54fconst 6558 . . . . 5 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6 ffn 6507 . . . . 5 (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
75, 6ax-mp 5 . . . 4 ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)
83, 7pm3.2i 473 . . 3 (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
9 disjdif 4419 . . 3 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10 fnun 6456 . . 3 (((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) ∧ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅) → (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
118, 9, 10mp2an 690 . 2 (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12 df-fullfun 33329 . . . 4 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
1312fneq1i 6443 . . 3 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14 unvdif 4421 . . . . 5 (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
1514eqcomi 2828 . . . 4 V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
1615fneq2i 6444 . . 3 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1713, 16bitri 277 . 2 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1811, 17mpbir 233 1 FullFun𝐹 Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1531  Vcvv 3493  cdif 3931  cun 3932  cin 3933  c0 4289  {csn 4559   × cxp 5546  dom cdm 5548  Fun wfun 6342   Fn wfn 6343  wf 6344  Funpartcfunpart 33303  FullFuncfullfn 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-symdif 4217  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7681  df-2nd 7682  df-txp 33308  df-singleton 33316  df-singles 33317  df-image 33318  df-funpart 33328  df-fullfun 33329
This theorem is referenced by:  brfullfun  33402
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