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Theorem fullfunfnv 35451
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 35448 . . . . 5 Fun Funpart𝐹
2 funfn 6572 . . . . 5 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
31, 2mpbi 229 . . . 4 Funpart𝐹 Fn dom Funpart𝐹
4 0ex 5300 . . . . . 6 ∅ ∈ V
54fconst 6771 . . . . 5 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6 ffn 6711 . . . . 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 4466 . . 3 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10 fnun 6657 . . 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 689 . 2 (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12 df-fullfun 35380 . . . 4 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
1312fneq1i 6640 . . 3 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14 unvdif 4469 . . . . 5 (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
1514eqcomi 2735 . . . 4 V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
1615fneq2i 6641 . . 3 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1713, 16bitri 275 . 2 (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
1811, 17mpbir 230 1 FullFun𝐹 Fn V
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  Vcvv 3468  cdif 3940  cun 3941  cin 3942  c0 4317  {csn 4623   × cxp 5667  dom cdm 5669  Fun wfun 6531   Fn wfn 6532  wf 6533  Funpartcfunpart 35354  FullFuncfullfn 35355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-symdif 4237  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7974  df-2nd 7975  df-txp 35359  df-singleton 35367  df-singles 35368  df-image 35369  df-funpart 35379  df-fullfun 35380
This theorem is referenced by:  brfullfun  35453
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