Theorem fullfunfv 36012

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 6828	. . . 4 ⊢ (𝑥 = 𝐴 → (FullFun𝐹‘𝑥) = (FullFun𝐹‘𝐴))
2		fveq2 6828	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
3	1, 2	eqeq12d 2749	. . 3 ⊢ (𝑥 = 𝐴 → ((FullFun𝐹‘𝑥) = (𝐹‘𝑥) ↔ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)))
4		df-fullfun 35938	. . . . 5 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
5	4	fveq1i 6829	. . . 4 ⊢ (FullFun𝐹‘𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6		disjdif 4421	. . . . . 6 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7		funpartfun 36008	. . . . . . . 8 ⊢ Fun Funpart𝐹
8		funfn 6516	. . . . . . . 8 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ Funpart𝐹 Fn dom Funpart𝐹
10		0ex 5247	. . . . . . . . 9 ⊢ ∅ ∈ V
11	10	fconst 6714	. . . . . . . 8 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12		ffn 6656	. . . . . . . 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 6919	. . . . . . 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 3441	. . . . . . . 8 ⊢ 𝑥 ∈ V
18		eldif 3908	. . . . . . . 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 6920	. . . . . . . . . 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 7144	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
24	22, 23	eqtrd 2768	. . . . . . 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 6860	. . . . . 6 ⊢ (¬ 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹‘𝑥) = ∅)
27	25, 26	eqtr4d 2771	. . . . 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 36010	. . . 4 ⊢ (Funpart𝐹‘𝑥) = (𝐹‘𝑥)
30	5, 28, 29	3eqtri 2760	. . 3 ⊢ (FullFun𝐹‘𝑥) = (𝐹‘𝑥)
31	3, 30	vtoclg 3508	. 2 ⊢ (𝐴 ∈ V → (FullFun𝐹‘𝐴) = (𝐹‘𝐴))
32		fvprc 6820	. . 3 ⊢ (¬ 𝐴 ∈ V → (FullFun𝐹‘𝐴) = ∅)
33		fvprc 6820	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
34	32, 33	eqtr4d 2771	. 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 2113  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897  ∅c0 4282  {csn 4575   × cxp 5617  dom cdm 5619  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  Funpartcfunpart 35912  FullFuncfullfn 35913

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-symdif 4202  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-1st 7927  df-2nd 7928  df-txp 35917  df-singleton 35925  df-singles 35926  df-image 35927  df-funpart 35937  df-fullfun 35938

This theorem is referenced by:  brfullfun  36013