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Theorem fullfunfnv 32558
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 32555 . . . . 5 Fun Funpart𝐹
2 funfn 6129 . . . . 5 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
31, 2mpbi 222 . . . 4 Funpart𝐹 Fn dom Funpart𝐹
4 0ex 4982 . . . . . 6 ∅ ∈ V
54fconst 6304 . . . . 5 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6 ffn 6254 . . . . 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 463 . . 3 (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
9 disjdif 4232 . . 3 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10 fnun 6206 . . 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 684 . 2 (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12 df-fullfun 32487 . . . 4 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
1312fneq1i 6194 . . 3 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14 unvdif 4234 . . . . 5 (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
1514eqcomi 2806 . . . 4 V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
1615fneq2i 6195 . . 3 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1713, 16bitri 267 . 2 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1811, 17mpbir 223 1 FullFun𝐹 Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  Vcvv 3383  cdif 3764  cun 3765  cin 3766  c0 4113  {csn 4366   × cxp 5308  dom cdm 5310  Fun wfun 6093   Fn wfn 6094  wf 6095  Funpartcfunpart 32461  FullFuncfullfn 32462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-symdif 4039  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-eprel 5223  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fo 6105  df-fv 6107  df-1st 7399  df-2nd 7400  df-txp 32466  df-singleton 32474  df-singles 32475  df-image 32476  df-funpart 32486  df-fullfun 32487
This theorem is referenced by:  brfullfun  32560
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