Theorem ssnnf1octb 45237

Description: There exists a bijection between a subset of ℕ and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Assertion

Ref	Expression
ssnnf1octb	⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))

Distinct variable group:   𝐴,𝑓

Proof of Theorem ssnnf1octb

Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnfoctb 45091	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑔 𝑔:ℕ–onto→𝐴)
2		fofn 6737	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
3		nnex 12131	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → ℕ ∈ V)
5		ltwenn 13869	. . . . . . 7 ⊢ < We ℕ
6	5	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → < We ℕ)
7	2, 4, 6	wessf1orn 45229	. . . . 5 ⊢ (𝑔:ℕ–onto→𝐴 → ∃𝑥 ∈ 𝒫 ℕ(𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
8		f1odm 6767	. . . . . . . . . . 11 ⊢ ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → dom (𝑔 ↾ 𝑥) = 𝑥)
9	8	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) = 𝑥)
10		elpwi 4557	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 ℕ → 𝑥 ⊆ ℕ)
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 ⊆ ℕ)
12	9, 11	eqsstrd 3969	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) ⊆ ℕ)
13	12	3adant1 1130	. . . . . . . 8 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) ⊆ ℕ)
14		simpr 484	. . . . . . . . . 10 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
15		eqidd 2732	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥) = (𝑔 ↾ 𝑥))
16	8	eqcomd 2737	. . . . . . . . . . . 12 ⊢ ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → 𝑥 = dom (𝑔 ↾ 𝑥))
17	16	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 = dom (𝑔 ↾ 𝑥))
18		forn 6738	. . . . . . . . . . . 12 ⊢ (𝑔:ℕ–onto→𝐴 → ran 𝑔 = 𝐴)
19	18	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → ran 𝑔 = 𝐴)
20	15, 17, 19	f1oeq123d 6757	. . . . . . . . . 10 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 ↔ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴))
21	14, 20	mpbid 232	. . . . . . . . 9 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴)
22	21	3adant2 1131	. . . . . . . 8 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴)
23		vex 3440	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
24	23	resex 5978	. . . . . . . . 9 ⊢ (𝑔 ↾ 𝑥) ∈ V
25		dmeq 5843	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → dom 𝑓 = dom (𝑔 ↾ 𝑥))
26	25	sseq1d 3966	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (dom 𝑓 ⊆ ℕ ↔ dom (𝑔 ↾ 𝑥) ⊆ ℕ))
27		id 22	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝑓 = (𝑔 ↾ 𝑥))
28		eqidd 2732	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝐴 = 𝐴)
29	27, 25, 28	f1oeq123d 6757	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴))
30	26, 29	anbi12d 632	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → ((dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴) ↔ (dom (𝑔 ↾ 𝑥) ⊆ ℕ ∧ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴)))
31	24, 30	spcev 3561	. . . . . . . 8 ⊢ ((dom (𝑔 ↾ 𝑥) ⊆ ℕ ∧ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))
32	13, 22, 31	syl2anc 584	. . . . . . 7 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))
33	32	3exp 1119	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → (𝑥 ∈ 𝒫 ℕ → ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))))
34	33	rexlimdv 3131	. . . . 5 ⊢ (𝑔:ℕ–onto→𝐴 → (∃𝑥 ∈ 𝒫 ℕ(𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴)))
35	7, 34	mpd 15	. . . 4 ⊢ (𝑔:ℕ–onto→𝐴 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))
36	35	a1i 11	. . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (𝑔:ℕ–onto→𝐴 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴)))
37	36	exlimdv 1934	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∃𝑔 𝑔:ℕ–onto→𝐴 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴)))
38	1, 37	mpd 15	1 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550   class class class wbr 5091   We wwe 5568  dom cdm 5616  ran crn 5617   ↾ cres 5618  –onto→wfo 6479  –1-1-onto→wf1o 6480  ωcom 7796   ≼ cdom 8867   < clt 11146  ℕcn 12125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733

This theorem is referenced by:  isomennd  46575