Theorem fullfunfv 34228

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 6768	. . . 4 ⊢ (𝑥 = 𝐴 → (FullFun𝐹‘𝑥) = (FullFun𝐹‘𝐴))
2		fveq2 6768	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	1, 2	eqeq12d 2755	. . 3 ⊢ (𝑥 = 𝐴 → ((FullFun𝐹‘𝑥) = (𝐹‘𝑥) ↔ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)))
4		df-fullfun 34156	. . . . 5 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
5	4	fveq1i 6769	. . . 4 ⊢ (FullFun𝐹‘𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6		disjdif 4410	. . . . . 6 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7		funpartfun 34224	. . . . . . . 8 ⊢ Fun Funpart𝐹
8		funfn 6460	. . . . . . . 8 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
9	7, 8	mpbi 229	. . . . . . 7 ⊢ Funpart𝐹 Fn dom Funpart𝐹
10		0ex 5234	. . . . . . . . 9 ⊢ ∅ ∈ V
11	10	fconst 6656	. . . . . . . 8 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12		ffn 6596	. . . . . . . 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 6853	. . . . . . 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 1449	. . . . . 6 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
16	6, 15	mpan 686	. . . . 5 ⊢ (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹‘𝑥))
17		vex 3434	. . . . . . . 8 ⊢ 𝑥 ∈ V
18		eldif 3901	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
19	17, 18	mpbiran 705	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20		fvun2 6854	. . . . . . . . . 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 1449	. . . . . . . . 9 ⊢ (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
22	6, 21	mpan 686	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
23	10	fvconst2 7073	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
24	22, 23	eqtrd 2779	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
25	19, 24	sylbir 234	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26		ndmfv 6798	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹‘𝑥) = ∅)
27	25, 26	eqtr4d 2782	. . . . 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 34226	. . . 4 ⊢ (Funpart𝐹‘𝑥) = (𝐹‘𝑥)
30	5, 28, 29	3eqtri 2771	. . 3 ⊢ (FullFun𝐹‘𝑥) = (𝐹‘𝑥)
31	3, 30	vtoclg 3503	. 2 ⊢ (𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
32		fvprc 6760	. . 3 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = ∅)
33		fvprc 6760	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
34	32, 33	eqtr4d 2782	. 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 2109  Vcvv 3430   ∖ cdif 3888   ∪ cun 3889   ∩ cin 3890  ∅c0 4261  {csn 4566   × cxp 5586  dom cdm 5588  Fun wfun 6424   Fn wfn 6425  ⟶wf 6426  ‘cfv 6430  Funpartcfunpart 34130  FullFuncfullfn 34131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-symdif 4181  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-eprel 5494  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-1st 7817  df-2nd 7818  df-txp 34135  df-singleton 34143  df-singles 34144  df-image 34145  df-funpart 34155  df-fullfun 34156

This theorem is referenced by:  brfullfun  34229