Theorem ennnfonelemr 12378

Description: Lemma for ennnfone 12380. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)

Hypotheses

Ref	Expression
ennnfonelemr.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemr.f	⊢ (𝜑 → 𝐹:ℕ0–onto→𝐴)
ennnfonelemr.n	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))

Assertion

Ref	Expression
ennnfonelemr	⊢ (𝜑 → 𝐴 ≈ ℕ)

Distinct variable groups:   𝑦,𝐴,𝑥   𝑛,𝐹,𝑗,𝑘

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗,𝑘,𝑛)   𝐴(𝑗,𝑘,𝑛)   𝐹(𝑥,𝑦)

Proof of Theorem ennnfonelemr

Dummy variables 𝑎 𝑏 𝑑 𝑒 𝑓 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemr.dceq	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		equequ1 1705	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
3	2	dcbid 833	. . . 4 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
4		equequ2 1706	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
5	4	dcbid 833	. . . 4 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
6	3, 5	cbvral2v 2709	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
7	1, 6	sylib 121	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
8		ennnfonelemr.f	. 2 ⊢ (𝜑 → 𝐹:ℕ0–onto→𝐴)
9		ennnfonelemr.n	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗))
10		fveq2 5496	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝐹‘𝑗) = (𝐹‘𝑓))
11	10	neeq2d 2359	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑓)))
12	11	cbvralv 2696	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
13	12	rexbii 2477	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
14		fveq2 5496	. . . . . . . . 9 ⊢ (𝑘 = 𝑒 → (𝐹‘𝑘) = (𝐹‘𝑒))
15	14	neeq1d 2358	. . . . . . . 8 ⊢ (𝑘 = 𝑒 → ((𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ (𝐹‘𝑒) ≠ (𝐹‘𝑓)))
16	15	ralbidv 2470	. . . . . . 7 ⊢ (𝑘 = 𝑒 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
17	16	cbvrexv 2697	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
18	13, 17	bitri 183	. . . . 5 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
19	18	ralbii 2476	. . . 4 ⊢ (∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑛 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
20		oveq2 5861	. . . . . . 7 ⊢ (𝑛 = 𝑑 → (0...𝑛) = (0...𝑑))
21	20	raleqdv 2671	. . . . . 6 ⊢ (𝑛 = 𝑑 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
22	21	rexbidv 2471	. . . . 5 ⊢ (𝑛 = 𝑑 → (∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
23	22	cbvralv 2696	. . . 4 ⊢ (∀𝑛 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
24	19, 23	bitri 183	. . 3 ⊢ (∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
25	9, 24	sylib 121	. 2 ⊢ (𝜑 → ∀𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
26		oveq1 5860	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑐 + 1) = (𝑎 + 1))
27	26	cbvmptv 4085	. . 3 ⊢ (𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
28		freceq1 6371	. . 3 ⊢ ((𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1)) → frec((𝑐 ∈ ℤ ↦ (𝑐 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0))
29	27, 28	ax-mp 5	. 2 ⊢ frec((𝑐 ∈ ℤ ↦ (𝑐 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0)
30	7, 8, 25, 29	ennnfonelemnn0 12377	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4  DECID wdc 829   = wceq 1348   ≠ wne 2340  ∀wral 2448  ∃wrex 2449   class class class wbr 3989   ↦ cmpt 4050  –onto→wfo 5196  ‘cfv 5198  (class class class)co 5853  freccfrec 6369   ≈ cen 6716  0cc0 7774  1c1 7775   + caddc 7777  ℕcn 8878  ℕ₀cn0 9135  ℤcz 9212  ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-er 6513  df-pm 6629  df-en 6719  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-seqfrec 10402

This theorem is referenced by:  ennnfone  12380