Theorem funpartfv 35927

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 35856	. . 3 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2	1	fveq1i 6908	. 2 ⊢ (Funpart𝐹‘𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3		fvres 6926	. . 3 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹‘𝐴))
4		nfvres 6948	. . . 4 ⊢ (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5		funpartlem 35924	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6		eusn 4735	. . . . . . . . 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 6109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
9	8	elvd 3484	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
10		df-br 5149	. . . . . . . . . 10 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
11	9, 10	bitr4di 289	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
12	11	eubidv 2584	. . . . . . . 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 6895	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
16	14, 15	biimtrdi 253	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹‘𝐴) = ∅))
17		fvprc 6899	. . . . . 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 2778	. . 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 2763	1 ⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∃!weu 2566  Vcvv 3478   ∩ cin 3962  ∅c0 4339  {csn 4631  ⟨cop 4637   class class class wbr 5148   × cxp 5687  dom cdm 5689   ↾ cres 5691   “ cima 5692   ∘ ccom 5693  ‘cfv 6563  Singletoncsingle 35820   Singletons csingles 35821  Imagecimage 35822  Funpartcfunpart 35831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-singleton 35844  df-singles 35845  df-image 35846  df-funpart 35856

This theorem is referenced by:  fullfunfv  35929