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Theorem fullfunfv 32382
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 6354 . . . 4 (𝑥 = 𝐴 → (FullFun𝐹𝑥) = (FullFun𝐹𝐴))
2 fveq2 6354 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31, 2eqeq12d 2776 . . 3 (𝑥 = 𝐴 → ((FullFun𝐹𝑥) = (𝐹𝑥) ↔ (FullFun𝐹𝐴) = (𝐹𝐴)))
4 df-fullfun 32310 . . . . 5 FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
54fveq1i 6355 . . . 4 (FullFun𝐹𝑥) = ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥)
6 disjdif 4185 . . . . . 6 (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅
7 funpartfun 32378 . . . . . . . 8 Fun Funpart𝐹
8 funfn 6080 . . . . . . . 8 (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹)
97, 8mpbi 220 . . . . . . 7 Funpart𝐹 Fn dom Funpart𝐹
10 0ex 4943 . . . . . . . . 9 ∅ ∈ V
1110fconst 6253 . . . . . . . 8 ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅}
12 ffn 6207 . . . . . . . 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 6433 . . . . . . 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 1563 . . . . . 6 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
166, 15mpan 708 . . . . 5 (𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
17 vex 3344 . . . . . . . 8 𝑥 ∈ V
18 eldif 3726 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom Funpart𝐹))
1917, 18mpbiran 991 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) ↔ ¬ 𝑥 ∈ dom Funpart𝐹)
20 fvun2 6434 . . . . . . . . . 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 1563 . . . . . . . . 9 (((dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ ∧ 𝑥 ∈ (V ∖ dom Funpart𝐹)) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
226, 21mpan 708 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (((V ∖ dom Funpart𝐹) × {∅})‘𝑥))
2310fvconst2 6635 . . . . . . . 8 (𝑥 ∈ (V ∖ dom Funpart𝐹) → (((V ∖ dom Funpart𝐹) × {∅})‘𝑥) = ∅)
2422, 23eqtrd 2795 . . . . . . 7 (𝑥 ∈ (V ∖ dom Funpart𝐹) → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
2519, 24sylbir 225 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = ∅)
26 ndmfv 6381 . . . . . 6 𝑥 ∈ dom Funpart𝐹 → (Funpart𝐹𝑥) = ∅)
2725, 26eqtr4d 2798 . . . . 5 𝑥 ∈ dom Funpart𝐹 → ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥))
2816, 27pm2.61i 176 . . . 4 ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))‘𝑥) = (Funpart𝐹𝑥)
29 funpartfv 32380 . . . 4 (Funpart𝐹𝑥) = (𝐹𝑥)
305, 28, 293eqtri 2787 . . 3 (FullFun𝐹𝑥) = (𝐹𝑥)
313, 30vtoclg 3407 . 2 (𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
32 fvprc 6348 . . 3 𝐴 ∈ V → (FullFun𝐹𝐴) = ∅)
33 fvprc 6348 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
3432, 33eqtr4d 2798 . 2 𝐴 ∈ V → (FullFun𝐹𝐴) = (𝐹𝐴))
3531, 34pm2.61i 176 1 (FullFun𝐹𝐴) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  cdif 3713  cun 3714  cin 3715  c0 4059  {csn 4322   × cxp 5265  dom cdm 5267  Fun wfun 6044   Fn wfn 6045  wf 6046  cfv 6050  Funpartcfunpart 32284  FullFuncfullfn 32285
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-symdif 3988  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-eprel 5180  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fo 6056  df-fv 6058  df-1st 7335  df-2nd 7336  df-txp 32289  df-singleton 32297  df-singles 32298  df-image 32299  df-funpart 32309  df-fullfun 32310
This theorem is referenced by:  brfullfun  32383
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