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	⊢ (φ → G ∈ Funs )
frec0.3	⊢ (φ → I ∈ 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.

Step	Hyp	Ref	Expression
1		peano1 4402	. . . . . 6 ⊢ 0c ∈ Nn
2		frec0.3	. . . . . 6 ⊢ (φ → I ∈ dom G)
3		opexg 4587	. . . . . 6 ⊢ ((0c ∈ Nn ∧ I ∈ dom G) → ⟨0c, I⟩ ∈ V)
4	1, 2, 3	sylancr 644	. . . . 5 ⊢ (φ → ⟨0c, I⟩ ∈ V)
5		snidg 3758	. . . . 5 ⊢ (⟨0c, I⟩ ∈ V → ⟨0c, I⟩ ∈ {⟨0c, I⟩})
6	4, 5	syl 15	. . . 4 ⊢ (φ → ⟨0c, I⟩ ∈ {⟨0c, I⟩})
7	6	orcd 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 ⊢ (φ → G ∈ Funs )
11		pprodexg 5837	. . . . 5 ⊢ (((x ∈ V ↦ (x +c 1c)) ∈ V ∧ G ∈ Funs ) → PProd ((x ∈ V ↦ (x +c 1c)), G) ∈ V)
12	9, 10, 11	sylancr 644	. . . 4 ⊢ (φ → PProd ((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))
15	13, 14	eqtri 2373	. . . . 5 ⊢ F = Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G))
16	15	clos1basesucg 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⟩)))
17	8, 12, 16	sylancr 644	. . 3 ⊢ (φ → (⟨0c, I⟩ ∈ F ↔ (⟨0c, I⟩ ∈ {⟨0c, I⟩} ∨ ∃y ∈ F y PProd ((x ∈ V ↦ (x +c 1c)), G)⟨0c, I⟩)))
18	7, 17	mpbird 223	. 2 ⊢ (φ → ⟨0c, I⟩ ∈ F)
19		frec0.4	. . . 4 ⊢ (φ → ran G ⊆ dom G)
20	13, 10, 2, 19	fnfrec 6320	. . 3 ⊢ (φ → F Fn Nn )
21		fnopfvb 5359	. . 3 ⊢ ((F Fn Nn ∧ 0c ∈ Nn ) → ((F ‘0c) = I ↔ ⟨0c, I⟩ ∈ F))
22	20, 1, 21	sylancl 643	. 2 ⊢ (φ → ((F ‘0c) = I ↔ ⟨0c, I⟩ ∈ F))
23	18, 22	mpbird 223	1 ⊢ (φ → (F ‘0c) = I)

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737  1_cc1c 4134   Nn cnnc 4373  0_cc0c 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)