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Theorem fullfunfnv 31223
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 31220 . . . . 5 Fun Funpart𝐹
2 funfn 5833 . . . . 5 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
31, 2mpbi 219 . . . 4 Funpart𝐹 Fn dom Funpart𝐹
4 0ex 4718 . . . . . 6 ∅ ∈ V
54fconst 6004 . . . . 5 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6 ffn 5958 . . . . 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 470 . . 3 (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
9 disjdif 3992 . . 3 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10 fnun 5911 . . 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 704 . 2 (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12 df-fullfun 31151 . . . 4 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
1312fneq1i 5899 . . 3 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14 unvdif 3994 . . . . 5 (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
1514eqcomi 2619 . . . 4 V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
1615fneq2i 5900 . . 3 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1713, 16bitri 263 . 2 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1811, 17mpbir 220 1 FullFun𝐹 Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  Vcvv 3173  cdif 3537  cun 3538  cin 3539  c0 3874  {csn 4125   × cxp 5036  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  wf 5800  Funpartcfunpart 31125  FullFuncfullfn 31126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139  df-image 31140  df-funpart 31150  df-fullfun 31151
This theorem is referenced by:  brfullfun  31225
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