Theorem funpartfv 35909

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 35838	. . 3 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2	1	fveq1i 6921	. 2 ⊢ (Funpart𝐹‘𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3		fvres 6939	. . 3 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
4		nfvres 6961	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5		funpartlem 35906	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6		eusn 4755	. . . . . . . . 9 ⊢ (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
7	5, 6	bitr4i 278	. . . . . . . 8 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}))
8		elimasng 6118	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
9	8	elvd 3494	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
10		df-br 5167	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
11	9, 10	bitr4di 289	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
12	11	eubidv 2589	. . . . . . . 8 ⊢ (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
13	7, 12	bitrid 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 6908	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
16	14, 15	biimtrdi 253	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
17		fvprc 6912	. . . . . 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 2783	. . 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 2768	1 ⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃!weu 2571  Vcvv 3488   ∩ cin 3975  ∅c0 4352  {csn 4648  ⟨cop 4654   class class class wbr 5166   × cxp 5698  dom cdm 5700   ↾ cres 5702   “ cima 5703   ∘ ccom 5704  ‘cfv 6573  Singletoncsingle 35802   Singletons csingles 35803  Imagecimage 35804  Funpartcfunpart 35813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-symdif 4272  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-1st 8030  df-2nd 8031  df-txp 35818  df-singleton 35826  df-singles 35827  df-image 35828  df-funpart 35838

This theorem is referenced by:  fullfunfv  35911