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Theorem fullfunfv 35911
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 6920 . . . 4 (𝑥 = 𝐴 → (FullFun𝐹𝑥) = (FullFun𝐹𝐴))
2 fveq2 6920 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31, 2eqeq12d 2756 . . 3 (𝑥 = 𝐴 → ((FullFun𝐹𝑥) = (𝐹𝑥) ↔ (FullFun𝐹𝐴) = (𝐹𝐴)))
4 df-fullfun 35839 . . . . 5 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
54fveq1i 6921 . . . 4 (FullFun𝐹𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6 disjdif 4495 . . . . . 6 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7 funpartfun 35907 . . . . . . . 8 Fun Funpart𝐹
8 funfn 6608 . . . . . . . 8 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
97, 8mpbi 230 . . . . . . 7 Funpart𝐹 Fn dom Funpart𝐹
10 0ex 5325 . . . . . . . . 9 ∅ ∈ V
1110fconst 6807 . . . . . . . 8 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12 ffn 6747 . . . . . . . 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 7013 . . . . . . 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 689 . . . . 5 (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
17 vex 3492 . . . . . . . 8 𝑥 ∈ V
18 eldif 3986 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
1917, 18mpbiran 708 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20 fvun2 7014 . . . . . . . . . 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 689 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
2310fvconst2 7241 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
2422, 23eqtrd 2780 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
2519, 24sylbir 235 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26 ndmfv 6955 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹𝑥) = ∅)
2725, 26eqtr4d 2783 . . . . 5 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
2816, 27pm2.61i 182 . . . 4 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥)
29 funpartfv 35909 . . . 4 (Funpart𝐹𝑥) = (𝐹𝑥)
305, 28, 293eqtri 2772 . . 3 (FullFun𝐹𝑥) = (𝐹𝑥)
313, 30vtoclg 3566 . 2 (𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
32 fvprc 6912 . . 3 𝐴 ∈ V → (FullFun𝐹𝐴) = ∅)
33 fvprc 6912 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
3432, 33eqtr4d 2783 . 2 𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
3531, 34pm2.61i 182 1 (FullFun𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cdif 3973  cun 3974  cin 3975  c0 4352  {csn 4648   × cxp 5698  dom cdm 5700  Fun wfun 6567   Fn wfn 6568  wf 6569  cfv 6573  Funpartcfunpart 35813  FullFuncfullfn 35814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-symdif 4272  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-1st 8030  df-2nd 8031  df-txp 35818  df-singleton 35826  df-singles 35827  df-image 35828  df-funpart 35838  df-fullfun 35839
This theorem is referenced by:  brfullfun  35912
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