Theorem frec2uzrand 10857

Description: Range of 𝐺 (see frec2uz0d 10851). (Contributed by Jim Kingdon, 17-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (𝜑 → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)

Assertion

Ref	Expression
frec2uzrand	⊢ (𝜑 → ran 𝐺 = (ℤ≥‘𝐶))

Distinct variable groups:   𝑥,𝐶   𝜑,𝑥

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem frec2uzrand

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frec2uz.1	. 2 ⊢ (𝜑 → 𝐶 ∈ ℤ)
2		zex 9658	. . . . . . . . . . 11 ⊢ ℤ ∈ V
3	2	mptex 5943	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
4		vex 2824	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
5	3, 4	fvex 5715	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
6	5	ax-gen 1502	. . . . . . . 8 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
7		frecfnom 6672	. . . . . . . 8 ⊢ ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 𝐶 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) Fn ω)
8	6, 7	mpan 428	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) Fn ω)
9		frec2uz.2	. . . . . . . 8 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
10	9	fneq1i 5475	. . . . . . 7 ⊢ (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) Fn ω)
11	8, 10	sylibr 134	. . . . . 6 ⊢ (𝐶 ∈ ℤ → 𝐺 Fn ω)
12		fvelrnb 5750	. . . . . 6 ⊢ (𝐺 Fn ω → (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦))
13	11, 12	syl 14	. . . . 5 ⊢ (𝐶 ∈ ℤ → (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦))
14		simpl 109	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → 𝐶 ∈ ℤ)
15		simpr 110	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → 𝑧 ∈ ω)
16	14, 9, 15	frec2uzuzd 10854	. . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶))
17		eleq1 2301	. . . . . . 7 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) ↔ 𝑦 ∈ (ℤ≥‘𝐶)))
18	16, 17	syl5ibcom 155	. . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → ((𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)))
19	18	rexlimdva 2668	. . . . 5 ⊢ (𝐶 ∈ ℤ → (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)))
20	13, 19	sylbid 150	. . . 4 ⊢ (𝐶 ∈ ℤ → (𝑦 ∈ ran 𝐺 → 𝑦 ∈ (ℤ≥‘𝐶)))
21		eleq1 2301	. . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤 ∈ ran 𝐺 ↔ 𝐶 ∈ ran 𝐺))
22		eleq1 2301	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ ran 𝐺 ↔ 𝑦 ∈ ran 𝐺))
23		eleq1 2301	. . . . 5 ⊢ (𝑤 = (𝑦 + 1) → (𝑤 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺))
24		id 19	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℤ)
25	24, 9	frec2uz0d 10851	. . . . . 6 ⊢ (𝐶 ∈ ℤ → (𝐺‘∅) = 𝐶)
26		peano1 4741	. . . . . . 7 ⊢ ∅ ∈ ω
27		fnfvelrn 5840	. . . . . . 7 ⊢ ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺)
28	11, 26, 27	sylancl 417	. . . . . 6 ⊢ (𝐶 ∈ ℤ → (𝐺‘∅) ∈ ran 𝐺)
29	25, 28	eqeltrrd 2316	. . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ran 𝐺)
30		eluzel2 9936	. . . . . 6 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝐶 ∈ ℤ)
31	14, 9, 15	frec2uzsucd 10853	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
32		oveq1 6092	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) + 1) = (𝑦 + 1))
33	31, 32	sylan9eq 2291	. . . . . . . . . 10 ⊢ (((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1))
34		peano2 4742	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
35		fnfvelrn 5840	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺)
36	11, 34, 35	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺)
37	36	adantr 276	. . . . . . . . . 10 ⊢ (((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺)
38	33, 37	eqeltrrd 2316	. . . . . . . . 9 ⊢ (((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) ∧ (𝐺‘𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺)
39	38	ex 115	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝑧 ∈ ω) → ((𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺))
40	39	rexlimdva 2668	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺))
41	13, 40	sylbid 150	. . . . . 6 ⊢ (𝐶 ∈ ℤ → (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺))
42	30, 41	syl 14	. . . . 5 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺))
43	21, 22, 23, 22, 29, 42	uzind4 9998	. . . 4 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝑦 ∈ ran 𝐺)
44	20, 43	impbid1 142	. . 3 ⊢ (𝐶 ∈ ℤ → (𝑦 ∈ ran 𝐺 ↔ 𝑦 ∈ (ℤ≥‘𝐶)))
45	44	eqrdv 2236	. 2 ⊢ (𝐶 ∈ ℤ → ran 𝐺 = (ℤ≥‘𝐶))
46	1, 45	syl 14	1 ⊢ (𝜑 → ran 𝐺 = (ℤ≥‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ∅c0 3520   ↦ cmpt 4192  suc csuc 4510  ωcom 4737  ran crn 4775   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  freccfrec 6661  1c1 8181   + caddc 8183  ℤcz 9649  ℤ_≥cuz 9931

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-recs 6576  df-frec 6662  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932

This theorem is used by:  frec2uzf1od  10858