Theorem funpartfv 36437

Description: The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funpartfv	⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Proof of Theorem funpartfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-funpart 36364	. . 3 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2	1	fveq1i 6882	. 2 ⊢ (Funpart𝐹‘𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3		fvres 6900	. . 3 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
4		nfvres 6919	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5		funpartlem 36434	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6		eusn 4696	. . . . . . . . 9 ⊢ (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
7	5, 6	bitr4i 281	. . . . . . . 8 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}))
8		elimasng 6091	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
9	8	elvd 3461	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
10		df-br 5110	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
11	9, 10	bitr4di 292	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
12	11	eubidv 2614	. . . . . . . 8 ⊢ (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
13	7, 12	bitrid 286	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝐴𝐹𝑥))
14	13	notbid 321	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ¬ ∃!𝑥 𝐴𝐹𝑥))
15		tz6.12-2 6868	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
16	14, 15	biimtrdi 256	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
17		fvprc 6873	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
18	17	a1d 26	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
19	16, 18	pm2.61i 184	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅)
20	4, 19	eqtr4d 2801	. . 3 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
21	3, 20	pm2.61i 184	. 2 ⊢ ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴)
22	2, 21	eqtri 2786	1 ⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃!weu 2596  Vcvv 3455   ∩ cin 3904  ∅c0 4286  {csn 4589  ⟨cop 4595   class class class wbr 5109   × cxp 5659  dom cdm 5661   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  ‘cfv 6536  Singletoncsingle 36328   Singletons csingles 36329  Imagecimage 36330  Funpartcfunpart 36339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7982  df-2nd 7983  df-txp 36344  df-singleton 36352  df-singles 36353  df-image 36354  df-funpart 36364

This theorem is referenced by:  fullfunfv  36439