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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-1 5  ax-2 6  ax-3 7  ax-mp 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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