Theorem ennnfonelemr 12449

Description: Lemma for ennnfone 12451. 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 1723	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
3	2	dcbid 839	. . . 4 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
4		equequ2 1724	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
5	4	dcbid 839	. . . 4 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
6	3, 5	cbvral2v 2731	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
7	1, 6	sylib 122	. 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 5531	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝐹‘𝑗) = (𝐹‘𝑓))
11	10	neeq2d 2379	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑓)))
12	11	cbvralv 2718	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
13	12	rexbii 2497	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
14		fveq2 5531	. . . . . . . . 9 ⊢ (𝑘 = 𝑒 → (𝐹‘𝑘) = (𝐹‘𝑒))
15	14	neeq1d 2378	. . . . . . . 8 ⊢ (𝑘 = 𝑒 → ((𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ (𝐹‘𝑒) ≠ (𝐹‘𝑓)))
16	15	ralbidv 2490	. . . . . . 7 ⊢ (𝑘 = 𝑒 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
17	16	cbvrexv 2719	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
18	13, 17	bitri 184	. . . . 5 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
19	18	ralbii 2496	. . . 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 5900	. . . . . . 7 ⊢ (𝑛 = 𝑑 → (0...𝑛) = (0...𝑑))
21	20	raleqdv 2692	. . . . . 6 ⊢ (𝑛 = 𝑑 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
22	21	rexbidv 2491	. . . . 5 ⊢ (𝑛 = 𝑑 → (∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
23	22	cbvralv 2718	. . . 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 184	. . 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 122	. 2 ⊢ (𝜑 → ∀𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓))
26		oveq1 5899	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑐 + 1) = (𝑎 + 1))
27	26	cbvmptv 4114	. . 3 ⊢ (𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
28		freceq1 6412	. . 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 12448	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4  DECID wdc 835   = wceq 1364   ≠ wne 2360  ∀wral 2468  ∃wrex 2469   class class class wbr 4018   ↦ cmpt 4079  –onto→wfo 5230  ‘cfv 5232  (class class class)co 5892  freccfrec 6410   ≈ cen 6757  0cc0 7831  1c1 7832   + caddc 7834  ℕcn 8939  ℕ₀cn0 9196  ℤcz 9273  ...cfz 10028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-ltadd 7947

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-er 6554  df-pm 6670  df-en 6760  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-n0 9197  df-z 9274  df-uz 9549  df-fz 10029  df-seqfrec 10466

This theorem is referenced by:  ennnfone  12451