Theorem funpartfv 32052

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 31981	. . 3 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2	1	fveq1i 6192	. 2 ⊢ (Funpart𝐹‘𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3		fvres 6207	. . 3 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
4		nfvres 6224	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5		funpartlem 32049	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6		eusn 4265	. . . . . . . . 9 ⊢ (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
7	5, 6	bitr4i 267	. . . . . . . 8 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}))
8		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
9		elimasng 5491	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
10	8, 9	mpan2 707	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
11		df-br 4654	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
12	10, 11	syl6bbr 278	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
13	12	eubidv 2490	. . . . . . . 8 ⊢ (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
14	7, 13	syl5bb 272	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝐴𝐹𝑥))
15	14	notbid 308	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ¬ ∃!𝑥 𝐴𝐹𝑥))
16		tz6.12-2 6182	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
17	15, 16	syl6bi 243	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
18		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
19	18	a1d 25	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
20	17, 19	pm2.61i 176	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅)
21	4, 20	eqtr4d 2659	. . 3 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
22	3, 21	pm2.61i 176	. 2 ⊢ ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴)
23	2, 22	eqtri 2644	1 ⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200   ∩ cin 3573  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  dom cdm 5114   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  ‘cfv 5888  Singletoncsingle 31945   Singletons csingles 31946  Imagecimage 31947  Funpartcfunpart 31956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970  df-image 31971  df-funpart 31981

This theorem is referenced by:  fullfunfv  32054