Theorem ssnnf1octb 45624

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 45479	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑔 𝑔:ℕ–onto→𝐴)
2		fofn 6754	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
3		nnex 12180	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → ℕ ∈ V)
5		ltwenn 13924	. . . . . . 7 ⊢ < We ℕ
6	5	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → < We ℕ)
7	2, 4, 6	wessf1orn 45616	. . . . 5 ⊢ (𝑔:ℕ–onto→𝐴 → ∃𝑥 ∈ 𝒫 ℕ(𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
8		f1odm 6784	. . . . . . . . . . 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 4548	. . . . . . . . . . 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 3956	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) ⊆ ℕ)
13	12	3adant1 1131	. . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥) = (𝑔 ↾ 𝑥))
16	8	eqcomd 2742	. . . . . . . . . . . 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 6755	. . . . . . . . . . . 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 6774	. . . . . . . . . 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 1132	. . . . . . . 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 3433	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
24	23	resex 5994	. . . . . . . . 9 ⊢ (𝑔 ↾ 𝑥) ∈ V
25		dmeq 5858	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → dom 𝑓 = dom (𝑔 ↾ 𝑥))
26	25	sseq1d 3953	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (dom 𝑓 ⊆ ℕ ↔ dom (𝑔 ↾ 𝑥) ⊆ ℕ))
27		id 22	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝑓 = (𝑔 ↾ 𝑥))
28		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝐴 = 𝐴)
29	27, 25, 28	f1oeq123d 6774	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴))
30	26, 29	anbi12d 633	. . . . . . . . 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 3548	. . . . . . . 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 585	. . . . . . 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 1120	. . . . . 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 3136	. . . . 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 1935	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541   class class class wbr 5085   We wwe 5583  dom cdm 5631  ran crn 5632   ↾ cres 5633  –onto→wfo 6496  –1-1-onto→wf1o 6497  ωcom 7817   ≼ cdom 8891   < clt 11179  ℕcn 12174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789

This theorem is referenced by:  isomennd  46959