Theorem fullfunfv 35935

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 6858	. . . 4 ⊢ (𝑥 = 𝐴 → (FullFun𝐹‘𝑥) = (FullFun𝐹‘𝐴))
2		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	1, 2	eqeq12d 2745	. . 3 ⊢ (𝑥 = 𝐴 → ((FullFun𝐹‘𝑥) = (𝐹‘𝑥) ↔ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)))
4		df-fullfun 35863	. . . . 5 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
5	4	fveq1i 6859	. . . 4 ⊢ (FullFun𝐹‘𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6		disjdif 4435	. . . . . 6 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7		funpartfun 35931	. . . . . . . 8 ⊢ Fun Funpart𝐹
8		funfn 6546	. . . . . . . 8 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ Funpart𝐹 Fn dom Funpart𝐹
10		0ex 5262	. . . . . . . . 9 ⊢ ∅ ∈ V
11	10	fconst 6746	. . . . . . . 8 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12		ffn 6688	. . . . . . . 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 6952	. . . . . . 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 3451	. . . . . . . 8 ⊢ 𝑥 ∈ V
18		eldif 3924	. . . . . . . 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 6953	. . . . . . . . . 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 7178	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
24	22, 23	eqtrd 2764	. . . . . . 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 6893	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹‘𝑥) = ∅)
27	25, 26	eqtr4d 2767	. . . . 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 35933	. . . 4 ⊢ (Funpart𝐹‘𝑥) = (𝐹‘𝑥)
30	5, 28, 29	3eqtri 2756	. . 3 ⊢ (FullFun𝐹‘𝑥) = (𝐹‘𝑥)
31	3, 30	vtoclg 3520	. 2 ⊢ (𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
32		fvprc 6850	. . 3 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = ∅)
33		fvprc 6850	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
34	32, 33	eqtr4d 2767	. 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 1540   ∈ wcel 2109  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  {csn 4589   × cxp 5636  dom cdm 5638  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  Funpartcfunpart 35837  FullFuncfullfn 35838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-symdif 4216  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842  df-singleton 35850  df-singles 35851  df-image 35852  df-funpart 35862  df-fullfun 35863

This theorem is referenced by:  brfullfun  35936