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Theorem frec0 6321
Description: Calculate the value of the finite recursive function generator at zero. (Contributed by Scott Fenton, 31-Jul-2019.)
Hypotheses
Ref Expression
frec0.1 F = FRec (G, I)
frec0.2 (φG Funs )
frec0.3 (φI dom G)
frec0.4 (φ → ran G dom G)
Assertion
Ref Expression
frec0 (φ → (F ‘0c) = I)

Proof of Theorem frec0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano1 4402 . . . . . 6 0c Nn
2 frec0.3 . . . . . 6 (φI dom G)
3 opexg 4587 . . . . . 6 ((0c Nn I dom G) → 0c, I V)
41, 2, 3sylancr 644 . . . . 5 (φ0c, I V)
5 snidg 3758 . . . . 5 (0c, I V → 0c, I {0c, I})
64, 5syl 15 . . . 4 (φ0c, I {0c, I})
76orcd 381 . . 3 (φ → (0c, I {0c, I} y F y PProd ((x V (x +c 1c)), G)0c, I))
8 snex 4111 . . . 4 {0c, I} V
9 csucex 6259 . . . . 5 (x V (x +c 1c)) V
10 frec0.2 . . . . 5 (φG Funs )
11 pprodexg 5837 . . . . 5 (((x V (x +c 1c)) V G Funs ) → PProd ((x V (x +c 1c)), G) V)
129, 10, 11sylancr 644 . . . 4 (φPProd ((x V (x +c 1c)), G) V)
13 frec0.1 . . . . . 6 F = FRec (G, I)
14 df-frec 6310 . . . . . 6 FRec (G, I) = Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G))
1513, 14eqtri 2373 . . . . 5 F = Clos1 ({0c, I}, PProd ((x V (x +c 1c)), G))
1615clos1basesucg 5884 . . . 4 (({0c, I} V PProd ((x V (x +c 1c)), G) V) → (0c, I F ↔ (0c, I {0c, I} y F y PProd ((x V (x +c 1c)), G)0c, I)))
178, 12, 16sylancr 644 . . 3 (φ → (0c, I F ↔ (0c, I {0c, I} y F y PProd ((x V (x +c 1c)), G)0c, I)))
187, 17mpbird 223 . 2 (φ0c, I F)
19 frec0.4 . . . 4 (φ → ran G dom G)
2013, 10, 2, 19fnfrec 6320 . . 3 (φF Fn Nn )
21 fnopfvb 5359 . . 3 ((F Fn Nn 0c Nn ) → ((F ‘0c) = I0c, I F))
2220, 1, 21sylancl 643 . 2 (φ → ((F ‘0c) = I0c, I F))
2318, 22mpbird 223 1 (φ → (F ‘0c) = I)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wo 357   = wceq 1642   wcel 1710  wrex 2615  Vcvv 2859   wss 3257  {csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375  cop 4561   class class class wbr 4639  dom cdm 4772  ran crn 4773   Fn wfn 4776  cfv 4781   cmpt 5651   PProd cpprod 5737   Funs cfuns 5759   Clos1 cclos1 5872   FRec cfrec 6309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-clos1 5873  df-frec 6310
This theorem is referenced by: (None)
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