Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  frsucmptn Structured version   Visualization version   GIF version

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  Ⅎwnfc 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
 Copyright terms: Public domain W3C validator