Theorem fullfunfv 35980

Description: The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fullfunfv	⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)

Proof of Theorem fullfunfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝐴 → (FullFun𝐹‘𝑥) = (FullFun𝐹‘𝐴))
2		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	1, 2	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐴 → ((FullFun𝐹‘𝑥) = (𝐹‘𝑥) ↔ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)))
4		df-fullfun 35908	. . . . 5 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
5	4	fveq1i 6823	. . . 4 ⊢ (FullFun𝐹‘𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6		disjdif 4422	. . . . . 6 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7		funpartfun 35976	. . . . . . . 8 ⊢ Fun Funpart𝐹
8		funfn 6511	. . . . . . . 8 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ Funpart𝐹 Fn dom Funpart𝐹
10		0ex 5245	. . . . . . . . 9 ⊢ ∅ ∈ V
11	10	fconst 6709	. . . . . . . 8 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12		ffn 6651	. . . . . . . 8 ⊢ (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)
14		fvun1 6913	. . . . . . 7 ⊢ ((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹) ∧ ((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
15	9, 13, 14	mp3an12 1453	. . . . . 6 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
16	6, 15	mpan 690	. . . . 5 ⊢ (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
17		vex 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
18		eldif 3912	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
19	17, 18	mpbiran 709	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20		fvun2 6914	. . . . . . . . . 10 ⊢ ((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹) ∧ ((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹))) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
21	9, 13, 20	mp3an12 1453	. . . . . . . . 9 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
22	6, 21	mpan 690	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
23	10	fvconst2 7138	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
24	22, 23	eqtrd 2766	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
25	19, 24	sylbir 235	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26		ndmfv 6854	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹‘𝑥) = ∅)
27	25, 26	eqtr4d 2769	. . . . 5 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
28	16, 27	pm2.61i 182	. . . 4 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥)
29		funpartfv 35978	. . . 4 ⊢ (Funpart𝐹‘𝑥) = (𝐹‘𝑥)
30	5, 28, 29	3eqtri 2758	. . 3 ⊢ (FullFun𝐹‘𝑥) = (𝐹‘𝑥)
31	3, 30	vtoclg 3509	. 2 ⊢ (𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
32		fvprc 6814	. . 3 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = ∅)
33		fvprc 6814	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
34	32, 33	eqtr4d 2769	. 2 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
35	31, 34	pm2.61i 182	1 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901  ∅c0 4283  {csn 4576   × cxp 5614  dom cdm 5616  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  Funpartcfunpart 35882  FullFuncfullfn 35883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4203  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-eprel 5516  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-txp 35887  df-singleton 35895  df-singles 35896  df-image 35897  df-funpart 35907  df-fullfun 35908

This theorem is referenced by:  brfullfun  35981