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Theorem fullfunfv 31224
 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 6103 . . . 4 (𝑥 = 𝐴 → (FullFun𝐹𝑥) = (FullFun𝐹𝐴))
2 fveq2 6103 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31, 2eqeq12d 2625 . . 3 (𝑥 = 𝐴 → ((FullFun𝐹𝑥) = (𝐹𝑥) ↔ (FullFun𝐹𝐴) = (𝐹𝐴)))
4 df-fullfun 31151 . . . . 5 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
54fveq1i 6104 . . . 4 (FullFun𝐹𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6 disjdif 3992 . . . . . 6 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7 funpartfun 31220 . . . . . . . 8 Fun Funpart𝐹
8 funfn 5833 . . . . . . . 8 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
97, 8mpbi 219 . . . . . . 7 Funpart𝐹 Fn dom Funpart𝐹
10 0ex 4718 . . . . . . . . 9 ∅ ∈ V
1110fconst 6004 . . . . . . . 8 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12 ffn 5958 . . . . . . . 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 6179 . . . . . . 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 1406 . . . . . 6 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
166, 15mpan 702 . . . . 5 (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
17 vex 3176 . . . . . . . 8 𝑥 ∈ V
18 eldif 3550 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
1917, 18mpbiran 955 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20 fvun2 6180 . . . . . . . . . 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 1406 . . . . . . . . 9 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
226, 21mpan 702 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
2310fvconst2 6374 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
2422, 23eqtrd 2644 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
2519, 24sylbir 224 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26 ndmfv 6128 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹𝑥) = ∅)
2725, 26eqtr4d 2647 . . . . 5 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
2816, 27pm2.61i 175 . . . 4 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥)
29 funpartfv 31222 . . . 4 (Funpart𝐹𝑥) = (𝐹𝑥)
305, 28, 293eqtri 2636 . . 3 (FullFun𝐹𝑥) = (𝐹𝑥)
313, 30vtoclg 3239 . 2 (𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
32 fvprc 6097 . . 3 𝐴 ∈ V → (FullFun𝐹𝐴) = ∅)
33 fvprc 6097 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
3432, 33eqtr4d 2647 . 2 𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
3531, 34pm2.61i 175 1 (FullFun𝐹𝐴) = (𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   × cxp 5036  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Funpartcfunpart 31125  FullFuncfullfn 31126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139  df-image 31140  df-funpart 31150  df-fullfun 31151 This theorem is referenced by:  brfullfun  31225
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