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Theorem funpartfv 31686
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 (Funpart𝐹𝐴) = (𝐹𝐴)

Proof of Theorem funpartfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-funpart 31614 . . 3 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
21fveq1i 6151 . 2 (Funpart𝐹𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3 fvres 6165 . . 3 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹𝐴))
4 nfvres 6182 . . . 4 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5 funpartlem 31683 . . . . . . . . 9 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6 eusn 4240 . . . . . . . . 9 (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
75, 6bitr4i 267 . . . . . . . 8 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}))
8 vex 3194 . . . . . . . . . . 11 𝑥 ∈ V
9 elimasng 5454 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
108, 9mpan2 706 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
11 df-br 4619 . . . . . . . . . 10 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
1210, 11syl6bbr 278 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
1312eubidv 2494 . . . . . . . 8 (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
147, 13syl5bb 272 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝐴𝐹𝑥))
1514notbid 308 . . . . . 6 (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ¬ ∃!𝑥 𝐴𝐹𝑥))
16 tz6.12-2 6141 . . . . . 6 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
1715, 16syl6bi 243 . . . . 5 (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹𝐴) = ∅))
18 fvprc 6144 . . . . . 6 𝐴 ∈ V → (𝐹𝐴) = ∅)
1918a1d 25 . . . . 5 𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹𝐴) = ∅))
2017, 19pm2.61i 176 . . . 4 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹𝐴) = ∅)
214, 20eqtr4d 2663 . . 3 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹𝐴))
223, 21pm2.61i 176 . 2 ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹𝐴)
232, 22eqtri 2648 1 (Funpart𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wex 1701  wcel 1992  ∃!weu 2474  Vcvv 3191  cin 3559  c0 3896  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  dom cdm 5079  cres 5081  cima 5082  ccom 5083  cfv 5850  Singletoncsingle 31578   Singletons csingles 31579  Imagecimage 31580  Funpartcfunpart 31589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856  df-fv 5858  df-1st 7116  df-2nd 7117  df-txp 31594  df-singleton 31602  df-singles 31603  df-image 31604  df-funpart 31614
This theorem is referenced by:  fullfunfv  31688
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