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Theorem frsucmpt 7518
Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
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
frsucmpt ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem frsucmpt
StepHypRef Expression
1 frsuc 7517 . . 3 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
2 frsucmpt.4 . . . 4 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
32fveq1i 6179 . . 3 (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
42fveq1i 6179 . . . 4 (𝐹𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
54fveq2i 6181 . . 3 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
61, 3, 53eqtr4g 2679 . 2 (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
7 fvex 6188 . . 3 (𝐹𝐵) ∈ V
8 nfmpt1 4738 . . . . . . . 8 𝑥(𝑥 ∈ V ↦ 𝐶)
9 frsucmpt.1 . . . . . . . 8 𝑥𝐴
108, 9nfrdg 7495 . . . . . . 7 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11 nfcv 2762 . . . . . . 7 𝑥ω
1210, 11nfres 5387 . . . . . 6 𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
132, 12nfcxfr 2760 . . . . 5 𝑥𝐹
14 frsucmpt.2 . . . . 5 𝑥𝐵
1513, 14nffv 6185 . . . 4 𝑥(𝐹𝐵)
16 frsucmpt.3 . . . 4 𝑥𝐷
17 frsucmpt.5 . . . 4 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
18 eqid 2620 . . . 4 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1915, 16, 17, 18fvmptf 6287 . . 3 (((𝐹𝐵) ∈ V ∧ 𝐷𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
207, 19mpan 705 . 2 (𝐷𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
216, 20sylan9eq 2674 1 ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wnfc 2749  Vcvv 3195  cmpt 4720  cres 5106  suc csuc 5713  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:  frsucmpt2  7520  dffi3  8322  axdclem  9326  trpredlem1  31701  trpredtr  31704  trpredmintr  31705  trpred0  31710  trpredrec  31712
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