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Theorem frsucmptn 7519
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 7518 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 6179 . 2 (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3 frfnom 7515 . . . . . 6 (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4 fndm 5978 . . . . . 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 2691 . . . 4 (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7 peano2b 7066 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8 frsuc 7517 . . . . . . . 8 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
91fveq1i 6179 . . . . . . . . 9 (𝐹𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
109fveq2i 6181 . . . . . . . 8 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
118, 10syl6eqr 2672 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
12 nfmpt1 4738 . . . . . . . . . . . 12 𝑥(𝑥 ∈ V ↦ 𝐶)
13 frsucmpt.1 . . . . . . . . . . . 12 𝑥𝐴
1412, 13nfrdg 7495 . . . . . . . . . . 11 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15 nfcv 2762 . . . . . . . . . . 11 𝑥ω
1614, 15nfres 5387 . . . . . . . . . 10 𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
171, 16nfcxfr 2760 . . . . . . . . 9 𝑥𝐹
18 frsucmpt.2 . . . . . . . . 9 𝑥𝐵
1917, 18nffv 6185 . . . . . . . 8 𝑥(𝐹𝐵)
20 frsucmpt.3 . . . . . . . 8 𝑥𝐷
21 frsucmpt.5 . . . . . . . 8 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
22 eqid 2620 . . . . . . . 8 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
2319, 20, 21, 22fvmptnf 6288 . . . . . . 7 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
2411, 23sylan9eqr 2676 . . . . . 6 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
2524ex 450 . . . . 5 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
267, 25syl5bir 233 . . . 4 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
276, 26syl5bi 232 . . 3 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
28 ndmfv 6205 . . 3 (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
2927, 28pm2.61d1 171 . 2 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
302, 29syl5eq 2666 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  wnfc 2749  Vcvv 3195  c0 3907  cmpt 4720  dom cdm 5104  cres 5106  suc csuc 5713   Fn wfn 5871  cfv 5876  ωcom 7050  reccrdg 7490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491
This theorem is referenced by:  trpredlem1  31701
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