Theorem ennnfonelemr 12879

Description: Lemma for ennnfone 12881. 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 1736	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
3	2	dcbid 840	. . . 4 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
4		equequ2 1737	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
5	4	dcbid 840	. . . 4 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
6	3, 5	cbvral2v 2752	. . 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 5594	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝐹‘𝑗) = (𝐹‘𝑓))
11	10	neeq2d 2396	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑘) ≠ (𝐹‘𝑓)))
12	11	cbvralv 2739	. . . . . . 7 ⊢ (∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
13	12	rexbii 2514	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓))
14		fveq2 5594	. . . . . . . . 9 ⊢ (𝑘 = 𝑒 → (𝐹‘𝑘) = (𝐹‘𝑒))
15	14	neeq1d 2395	. . . . . . . 8 ⊢ (𝑘 = 𝑒 → ((𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ (𝐹‘𝑒) ≠ (𝐹‘𝑓)))
16	15	ralbidv 2507	. . . . . . 7 ⊢ (𝑘 = 𝑒 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑘) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
17	16	cbvrexv 2740	. . . . . 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 2513	. . . 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 5970	. . . . . . 7 ⊢ (𝑛 = 𝑑 → (0...𝑛) = (0...𝑑))
21	20	raleqdv 2709	. . . . . 6 ⊢ (𝑛 = 𝑑 → (∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
22	21	rexbidv 2508	. . . . 5 ⊢ (𝑛 = 𝑑 → (∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑛)(𝐹‘𝑒) ≠ (𝐹‘𝑓) ↔ ∃𝑒 ∈ ℕ0 ∀𝑓 ∈ (0...𝑑)(𝐹‘𝑒) ≠ (𝐹‘𝑓)))
23	22	cbvralv 2739	. . . 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 5969	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑐 + 1) = (𝑎 + 1))
27	26	cbvmptv 4151	. . 3 ⊢ (𝑐 ∈ ℤ ↦ (𝑐 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
28		freceq1 6496	. . 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 12878	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Syntax hints:   → wi 4  DECID wdc 836   = wceq 1373   ≠ wne 2377  ∀wral 2485  ∃wrex 2486   class class class wbr 4054   ↦ cmpt 4116  –onto→wfo 5283  ‘cfv 5285  (class class class)co 5962  freccfrec 6494   ≈ cen 6843  0cc0 7955  1c1 7956   + caddc 7958  ℕcn 9066  ℕ₀cn0 9325  ℤcz 9402  ...cfz 10160

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-er 6638  df-pm 6756  df-en 6846  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-fz 10161  df-seqfrec 10625

This theorem is referenced by:  ennnfone  12881