Theorem fullfunfnv 35928

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

Step	Hyp	Ref	Expression
1		funpartfun 35925	. . . . 5 ⊢ Fun Funpart𝐹
2		funfn 6598	. . . . 5 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
3	1, 2	mpbi 230	. . . 4 ⊢ Funpart𝐹 Fn dom Funpart𝐹
4		0ex 5313	. . . . . 6 ⊢ ∅ ∈ V
5	4	fconst 6795	. . . . 5 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
6		ffn 6737	. . . . 5 ⊢ (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
7	5, 6	ax-mp 5	. . . 4 ⊢ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)
8	3, 7	pm3.2i 470	. . 3 ⊢ (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
9		disjdif 4478	. . 3 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
10		fnun 6683	. . 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)))
11	8, 9, 10	mp2an 692	. 2 ⊢ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
12		df-fullfun 35857	. . . 4 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
13	12	fneq1i 6666	. . 3 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V)
14		unvdif 4481	. . . . 5 ⊢ (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V
15	14	eqcomi 2744	. . . 4 ⊢ V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))
16	15	fneq2i 6667	. . 3 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
17	13, 16	bitri 275	. 2 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)))
18	11, 17	mpbir 231	1 ⊢ FullFun𝐹 Fn V

Syntax hints:   ∧ wa 395   = wceq 1537  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  {csn 4631   × cxp 5687  dom cdm 5689  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  Funpartcfunpart 35831  FullFuncfullfn 35832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-singleton 35844  df-singles 35845  df-image 35846  df-funpart 35856  df-fullfun 35857

This theorem is referenced by:  brfullfun  35930