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Theorem frsucmpt2 8480
Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
frsucmpt2.1 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
frsucmpt2.2 (𝑦 = 𝑥𝐸 = 𝐶)
frsucmpt2.3 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
Assertion
Ref Expression
frsucmpt2 ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝐷   𝑥,𝐸
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐸(𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem frsucmpt2
StepHypRef Expression
1 nfcv 2905 . 2 𝑦𝐴
2 nfcv 2905 . 2 𝑦𝐵
3 nfcv 2905 . 2 𝑦𝐷
4 frsucmpt2.1 . . 3 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
5 frsucmpt2.2 . . . . . 6 (𝑦 = 𝑥𝐸 = 𝐶)
65cbvmptv 5255 . . . . 5 (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7 rdgeq1 8451 . . . . 5 ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
86, 7ax-mp 5 . . . 4 rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
98reseq1i 5993 . . 3 (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
104, 9eqtr4i 2768 . 2 𝐹 = (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω)
11 frsucmpt2.3 . 2 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
121, 2, 3, 10, 11frsucmpt 8478 1 ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  cres 5687  suc csuc 6386  cfv 6561  ωcom 7887  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  unblem2  9329  unblem3  9330  inf0  9661  trcl  9768  hsmexlem8  10464  wunex2  10778  wuncval2  10787  peano5nni  12269  peano2nn  12278  om2uzsuci  13989  noseqp1  28297  noseqind  28298  om2noseqsuc  28303  dfnns2  28362  neibastop2lem  36361
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