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Theorem funpartfv 32358
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 32287 . . 3 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
21fveq1i 6353 . 2 (Funpart𝐹𝐴) = ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴)
3 fvres 6368 . . 3 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = (𝐹𝐴))
4 nfvres 6385 . . . 4 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → ((𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))‘𝐴) = ∅)
5 funpartlem 32355 . . . . . . . . 9 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
6 eusn 4409 . . . . . . . . 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 3343 . . . . . . . . . . 11 𝑥 ∈ V
9 elimasng 5649 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
108, 9mpan2 709 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
11 df-br 4805 . . . . . . . . . 10 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
1210, 11syl6bbr 278 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
1312eubidv 2627 . . . . . . . 8 (𝐴 ∈ V → (∃!𝑥 𝑥 ∈ (𝐹 “ {𝐴}) ↔ ∃!𝑥 𝐴𝐹𝑥))
147, 13syl5bb 272 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃!𝑥 𝐴𝐹𝑥))
1514notbid 307 . . . . . 6 (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ¬ ∃!𝑥 𝐴𝐹𝑥))
16 tz6.12-2 6343 . . . . . 6 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
1715, 16syl6bi 243 . . . . 5 (𝐴 ∈ V → (¬ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → (𝐹𝐴) = ∅))
18 fvprc 6346 . . . . . 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 2797 . . 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 2782 1 (Funpart𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  Vcvv 3340  cin 3714  c0 4058  {csn 4321  cop 4327   class class class wbr 4804   × cxp 5264  dom cdm 5266  cres 5268  cima 5269  ccom 5270  cfv 6049  Singletoncsingle 32251   Singletons csingles 32252  Imagecimage 32253  Funpartcfunpart 32262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-symdif 3987  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-eprel 5179  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334  df-txp 32267  df-singleton 32275  df-singles 32276  df-image 32277  df-funpart 32287
This theorem is referenced by:  fullfunfv  32360
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