Theorem ennnfonelemr 12111

Description: Lemma for ennnfone 12113. 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 1689	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
3	2	dcbid 824	. . . 4 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
4		equequ2 1690	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
5	4	dcbid 824	. . . 4 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
6	3, 5	cbvral2v 2688	. . 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 5461	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝐹‘𝑗) = (𝐹‘𝑓))
11	10	neeq2d 2343	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑓)))
12	11	cbvralv 2677	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
13	12	rexbii 2461	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
14		fveq2 5461	. . . . . . . . 9 ⊢ (𝑘 = 𝑒 → (𝐹‘𝑘) = (𝐹‘𝑒))
15	14	neeq1d 2342	. . . . . . . 8 ⊢ (𝑘 = 𝑒 → ((𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ (𝐹‘𝑒) ≠ (𝐹‘𝑓)))
16	15	ralbidv 2454	. . . . . . 7 ⊢ (𝑘 = 𝑒 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
17	16	cbvrexv 2678	. . . . . 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 2460	. . . 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 5822	. . . . . . 7 ⊢ (𝑛 = 𝑑 → (0...𝑛) = (0...𝑑))
21	20	raleqdv 2655	. . . . . 6 ⊢ (𝑛 = 𝑑 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
22	21	rexbidv 2455	. . . . 5 ⊢ (𝑛 = 𝑑 → (∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
23	22	cbvralv 2677	. . . 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 5821	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑐 + 1) = (𝑎 + 1))
27	26	cbvmptv 4056	. . 3 ⊢ (𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
28		freceq1 6329	. . 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 12110	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4  DECID wdc 820   = wceq 1332   ≠ wne 2324  ∀wral 2432  ∃wrex 2433   class class class wbr 3961   ↦ cmpt 4021  –onto→wfo 5161  ‘cfv 5163  (class class class)co 5814  freccfrec 6327   ≈ cen 6672  0cc0 7711  1c1 7712   + caddc 7714  ℕcn 8812  ℕ₀cn0 9069  ℤcz 9146  ...cfz 9890

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-er 6469  df-pm 6585  df-en 6675  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147  df-uz 9419  df-fz 9891  df-seqfrec 10323

This theorem is referenced by:  ennnfone  12113