Theorem fullfunfv 36310

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 6871	. . . 4 ⊢ (𝑥 = 𝐴 → (FullFun𝐹‘𝑥) = (FullFun𝐹‘𝐴))
2		fveq2 6871	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	1, 2	eqeq12d 2781	. . 3 ⊢ (𝑥 = 𝐴 → ((FullFun𝐹‘𝑥) = (𝐹‘𝑥) ↔ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)))
4		df-fullfun 36236	. . . . 5 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
5	4	fveq1i 6872	. . . 4 ⊢ (FullFun𝐹‘𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6		disjdif 4429	. . . . . 6 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7		funpartfun 36306	. . . . . . . 8 ⊢ Fun Funpart𝐹
8		funfn 6555	. . . . . . . 8 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
9	7, 8	mpbi 233	. . . . . . 7 ⊢ Funpart𝐹 Fn dom Funpart𝐹
10		0ex 5262	. . . . . . . . 9 ⊢ ∅ ∈ V
11	10	fconst 6754	. . . . . . . 8 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12		ffn 6695	. . . . . . . 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 6962	. . . . . . 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 1475	. . . . . 6 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
16	6, 15	mpan 702	. . . . 5 ⊢ (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
17		vex 3461	. . . . . . . 8 ⊢ 𝑥 ∈ V
18		eldif 3917	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
19	17, 18	mpbiran 721	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20		fvun2 6963	. . . . . . . . . 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 1475	. . . . . . . . 9 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
22	6, 21	mpan 702	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
23	10	fvconst2 7192	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
24	22, 23	eqtrd 2800	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
25	19, 24	sylbir 238	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26		ndmfv 6903	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹‘𝑥) = ∅)
27	25, 26	eqtr4d 2803	. . . . 5 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
28	16, 27	pm2.61i 184	. . . 4 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥)
29		funpartfv 36308	. . . 4 ⊢ (Funpart𝐹‘𝑥) = (𝐹‘𝑥)
30	5, 28, 29	3eqtri 2792	. . 3 ⊢ (FullFun𝐹‘𝑥) = (𝐹‘𝑥)
31	3, 30	vtoclg 3525	. 2 ⊢ (𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
32		fvprc 6863	. . 3 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = ∅)
33		fvprc 6863	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
34	32, 33	eqtr4d 2803	. 2 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
35	31, 34	pm2.61i 184	1 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906  ∅c0 4288  {csn 4585   × cxp 5650  dom cdm 5652  Fun wfun 6519   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  Funpartcfunpart 36210  FullFuncfullfn 36211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-symdif 4208  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-1st 7974  df-2nd 7975  df-txp 36215  df-singleton 36223  df-singles 36224  df-image 36225  df-funpart 36235  df-fullfun 36236

This theorem is referenced by:  brfullfun  36311