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 31749
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 6158 . . . 4 (𝑥 = 𝐴 → (FullFun𝐹𝑥) = (FullFun𝐹𝐴))
2 fveq2 6158 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31, 2eqeq12d 2636 . . 3 (𝑥 = 𝐴 → ((FullFun𝐹𝑥) = (𝐹𝑥) ↔ (FullFun𝐹𝐴) = (𝐹𝐴)))
4 df-fullfun 31676 . . . . 5 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
54fveq1i 6159 . . . 4 (FullFun𝐹𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6 disjdif 4018 . . . . . 6 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7 funpartfun 31745 . . . . . . . 8 Fun Funpart𝐹
8 funfn 5887 . . . . . . . 8 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
97, 8mpbi 220 . . . . . . 7 Funpart𝐹 Fn dom Funpart𝐹
10 0ex 4760 . . . . . . . . 9 ∅ ∈ V
1110fconst 6058 . . . . . . . 8 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12 ffn 6012 . . . . . . . 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 6236 . . . . . . 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 1411 . . . . . 6 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
166, 15mpan 705 . . . . 5 (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
17 vex 3193 . . . . . . . 8 𝑥 ∈ V
18 eldif 3570 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
1917, 18mpbiran 952 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20 fvun2 6237 . . . . . . . . . 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 1411 . . . . . . . . 9 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
226, 21mpan 705 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
2310fvconst2 6434 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
2422, 23eqtrd 2655 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
2519, 24sylbir 225 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26 ndmfv 6185 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹𝑥) = ∅)
2725, 26eqtr4d 2658 . . . . 5 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
2816, 27pm2.61i 176 . . . 4 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥)
29 funpartfv 31747 . . . 4 (Funpart𝐹𝑥) = (𝐹𝑥)
305, 28, 293eqtri 2647 . . 3 (FullFun𝐹𝑥) = (𝐹𝑥)
313, 30vtoclg 3256 . 2 (𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
32 fvprc 6152 . . 3 𝐴 ∈ V → (FullFun𝐹𝐴) = ∅)
33 fvprc 6152 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
3432, 33eqtr4d 2658 . 2 𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
3531, 34pm2.61i 176 1 (FullFun𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  cun 3558  cin 3559  c0 3897  {csn 4155   × cxp 5082  dom cdm 5084  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  Funpartcfunpart 31650  FullFuncfullfn 31651
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3828  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-eprel 4995  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-singleton 31663  df-singles 31664  df-image 31665  df-funpart 31675  df-fullfun 31676
This theorem is referenced by:  brfullfun  31750
  Copyright terms: Public domain W3C validator