Theorem funpartfv 34247

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 34176	. . 3 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2	1	fveq1i 6775	. 2 ⊢ (Funpart𝐹‘𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3		fvres 6793	. . 3 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
4		nfvres 6810	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5		funpartlem 34244	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6		eusn 4666	. . . . . . . . 9 ⊢ (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
7	5, 6	bitr4i 277	. . . . . . . 8 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}))
8		elimasng 5996	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
9	8	elvd 3439	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
10		df-br 5075	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
11	9, 10	bitr4di 289	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
12	11	eubidv 2586	. . . . . . . 8 ⊢ (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
13	7, 12	syl5bb 283	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝐴𝐹𝑥))
14	13	notbid 318	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ¬ ∃!𝑥 𝐴𝐹𝑥))
15		tz6.12-2 6762	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
16	14, 15	syl6bi 252	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
17		fvprc 6766	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
18	17	a1d 25	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
19	16, 18	pm2.61i 182	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅)
20	4, 19	eqtr4d 2781	. . 3 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
21	3, 20	pm2.61i 182	. 2 ⊢ ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴)
22	2, 21	eqtri 2766	1 ⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432   ∩ cin 3886  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074   × cxp 5587  dom cdm 5589   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  ‘cfv 6433  Singletoncsingle 34140   Singletons csingles 34141  Imagecimage 34142  Funpartcfunpart 34151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-symdif 4176  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-singleton 34164  df-singles 34165  df-image 34166  df-funpart 34176

This theorem is referenced by:  fullfunfv  34249