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Theorem frsucmptn 7579
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 7578 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 6230 . 2 (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3 frfnom 7575 . . . . . 6 (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4 fndm 6028 . . . . . 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 2722 . . . 4 (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7 peano2b 7123 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8 frsuc 7577 . . . . . . . 8 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
91fveq1i 6230 . . . . . . . . 9 (𝐹𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
109fveq2i 6232 . . . . . . . 8 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
118, 10syl6eqr 2703 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
12 nfmpt1 4780 . . . . . . . . . . . 12 𝑥(𝑥 ∈ V ↦ 𝐶)
13 frsucmpt.1 . . . . . . . . . . . 12 𝑥𝐴
1412, 13nfrdg 7555 . . . . . . . . . . 11 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15 nfcv 2793 . . . . . . . . . . 11 𝑥ω
1614, 15nfres 5430 . . . . . . . . . 10 𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
171, 16nfcxfr 2791 . . . . . . . . 9 𝑥𝐹
18 frsucmpt.2 . . . . . . . . 9 𝑥𝐵
1917, 18nffv 6236 . . . . . . . 8 𝑥(𝐹𝐵)
20 frsucmpt.3 . . . . . . . 8 𝑥𝐷
21 frsucmpt.5 . . . . . . . 8 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
22 eqid 2651 . . . . . . . 8 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
2319, 20, 21, 22fvmptnf 6341 . . . . . . 7 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
2411, 23sylan9eqr 2707 . . . . . 6 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
2524ex 449 . . . . 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 6256 . . 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 2697 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  wnfc 2780  Vcvv 3231  c0 3948  cmpt 4762  dom cdm 5143  cres 5145  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  trpredlem1  31851
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