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Theorem frsucmptn 8378
Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 8377 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
frsucmpt.1 𝑥𝐴
frsucmpt.2 𝑥𝐵
frsucmpt.3 𝑥𝐷
frsucmpt.4 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
frsucmpt.5 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
frsucmptn 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Proof of Theorem frsucmptn
StepHypRef Expression
1 frsucmpt.4 . . 3 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
21fveq1i 6841 . 2 (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3 frfnom 8374 . . . . . 6 (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4 fndm 6603 . . . . . 6 ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω → dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω)
53, 4ax-mp 5 . . . . 5 dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω
65eleq2i 2830 . . . 4 (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7 peano2b 7812 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8 frsuc 8376 . . . . . . . 8 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
91fveq1i 6841 . . . . . . . . 9 (𝐹𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
109fveq2i 6843 . . . . . . . 8 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
118, 10eqtr4di 2796 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
12 nfmpt1 5212 . . . . . . . . . . . 12 𝑥(𝑥 ∈ V ↦ 𝐶)
13 frsucmpt.1 . . . . . . . . . . . 12 𝑥𝐴
1412, 13nfrdg 8353 . . . . . . . . . . 11 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15 nfcv 2906 . . . . . . . . . . 11 𝑥ω
1614, 15nfres 5938 . . . . . . . . . 10 𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
171, 16nfcxfr 2904 . . . . . . . . 9 𝑥𝐹
18 frsucmpt.2 . . . . . . . . 9 𝑥𝐵
1917, 18nffv 6850 . . . . . . . 8 𝑥(𝐹𝐵)
20 frsucmpt.3 . . . . . . . 8 𝑥𝐷
21 frsucmpt.5 . . . . . . . 8 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
22 eqid 2738 . . . . . . . 8 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
2319, 20, 21, 22fvmptnf 6968 . . . . . . 7 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
2411, 23sylan9eqr 2800 . . . . . 6 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
2524ex 414 . . . . 5 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
267, 25biimtrrid 242 . . . 4 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
276, 26biimtrid 241 . . 3 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
28 ndmfv 6875 . . 3 (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
2927, 28pm2.61d1 180 . 2 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
302, 29eqtrid 2790 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wnfc 2886  Vcvv 3444  c0 4281  cmpt 5187  dom cdm 5632  cres 5634  suc csuc 6318   Fn wfn 6489  cfv 6494  ωcom 7795  reccrdg 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-om 7796  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349
This theorem is referenced by: (None)
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