Theorem ennnfonelemr 11972

Description: Lemma for ennnfone 11974. 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 2668	. . 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 5429	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝐹‘𝑗) = (𝐹‘𝑓))
11	10	neeq2d 2328	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑓)))
12	11	cbvralv 2657	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
13	12	rexbii 2445	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
14		fveq2 5429	. . . . . . . . 9 ⊢ (𝑘 = 𝑒 → (𝐹‘𝑘) = (𝐹‘𝑒))
15	14	neeq1d 2327	. . . . . . . 8 ⊢ (𝑘 = 𝑒 → ((𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ (𝐹‘𝑒) ≠ (𝐹‘𝑓)))
16	15	ralbidv 2438	. . . . . . 7 ⊢ (𝑘 = 𝑒 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
17	16	cbvrexv 2658	. . . . . 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 2444	. . . 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 5790	. . . . . . 7 ⊢ (𝑛 = 𝑑 → (0...𝑛) = (0...𝑑))
21	20	raleqdv 2635	. . . . . 6 ⊢ (𝑛 = 𝑑 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
22	21	rexbidv 2439	. . . . 5 ⊢ (𝑛 = 𝑑 → (∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
23	22	cbvralv 2657	. . . 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 5789	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑐 + 1) = (𝑎 + 1))
27	26	cbvmptv 4032	. . 3 ⊢ (𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
28		freceq1 6297	. . 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 11971	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4  DECID wdc 820   = wceq 1332   ≠ wne 2309  ∀wral 2417  ∃wrex 2418   class class class wbr 3937   ↦ cmpt 3997  –onto→wfo 5129  ‘cfv 5131  (class class class)co 5782  freccfrec 6295   ≈ cen 6640  0cc0 7644  1c1 7645   + caddc 7647  ℕcn 8744  ℕ₀cn0 9001  ℤcz 9078  ...cfz 9821

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

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 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-er 6437  df-pm 6553  df-en 6643  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-seqfrec 10250

This theorem is referenced by:  ennnfone  11974