Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fullfunfv Structured version   Visualization version   GIF version

Theorem fullfunfv 34988
Description: The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fullfunfv (FullFun𝐹𝐴) = (𝐹𝐴)

Proof of Theorem fullfunfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . 4 (𝑥 = 𝐴 → (FullFun𝐹𝑥) = (FullFun𝐹𝐴))
2 fveq2 6891 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31, 2eqeq12d 2748 . . 3 (𝑥 = 𝐴 → ((FullFun𝐹𝑥) = (𝐹𝑥) ↔ (FullFun𝐹𝐴) = (𝐹𝐴)))
4 df-fullfun 34916 . . . . 5 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
54fveq1i 6892 . . . 4 (FullFun𝐹𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6 disjdif 4471 . . . . . 6 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7 funpartfun 34984 . . . . . . . 8 Fun Funpart𝐹
8 funfn 6578 . . . . . . . 8 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
97, 8mpbi 229 . . . . . . 7 Funpart𝐹 Fn dom Funpart𝐹
10 0ex 5307 . . . . . . . . 9 ∅ ∈ V
1110fconst 6777 . . . . . . . 8 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12 ffn 6717 . . . . . . . 8 (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹))
1311, 12ax-mp 5 . . . . . . 7 ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)
14 fvun1 6982 . . . . . . 7 ((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹) ∧ ((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
159, 13, 14mp3an12 1451 . . . . . 6 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
166, 15mpan 688 . . . . 5 (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
17 vex 3478 . . . . . . . 8 𝑥 ∈ V
18 eldif 3958 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
1917, 18mpbiran 707 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20 fvun2 6983 . . . . . . . . . 10 ((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹) ∧ ((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹))) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
219, 13, 20mp3an12 1451 . . . . . . . . 9 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
226, 21mpan 688 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
2310fvconst2 7207 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
2422, 23eqtrd 2772 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
2519, 24sylbir 234 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26 ndmfv 6926 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹𝑥) = ∅)
2725, 26eqtr4d 2775 . . . . 5 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
2816, 27pm2.61i 182 . . . 4 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥)
29 funpartfv 34986 . . . 4 (Funpart𝐹𝑥) = (𝐹𝑥)
305, 28, 293eqtri 2764 . . 3 (FullFun𝐹𝑥) = (𝐹𝑥)
313, 30vtoclg 3556 . 2 (𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
32 fvprc 6883 . . 3 𝐴 ∈ V → (FullFun𝐹𝐴) = ∅)
33 fvprc 6883 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
3432, 33eqtr4d 2775 . 2 𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
3531, 34pm2.61i 182 1 (FullFun𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945  cun 3946  cin 3947  c0 4322  {csn 4628   × cxp 5674  dom cdm 5676  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  Funpartcfunpart 34890  FullFuncfullfn 34891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977  df-2nd 7978  df-txp 34895  df-singleton 34903  df-singles 34904  df-image 34905  df-funpart 34915  df-fullfun 34916
This theorem is referenced by:  brfullfun  34989
  Copyright terms: Public domain W3C validator