Theorem ssnnf1octb 45771

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 45627	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑔 𝑔:ℕ–onto→𝐴)
2		fofn 6784	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
3		nnex 12227	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → ℕ ∈ V)
5		ltwenn 13986	. . . . . . 7 ⊢ < We ℕ
6	5	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → < We ℕ)
7	2, 4, 6	wessf1orn 45763	. . . . 5 ⊢ (𝑔:ℕ–onto→𝐴 → ∃𝑥 ∈ 𝒫 ℕ(𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
8		f1odm 6814	. . . . . . . . . . 11 ⊢ ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → dom (𝑔 ↾ 𝑥) = 𝑥)
9	8	adantl 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) = 𝑥)
10		elpwi 4565	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 ℕ → 𝑥 ⊆ ℕ)
11	10	adantr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 ⊆ ℕ)
12	9, 11	eqsstrd 3973	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) ⊆ ℕ)
13	12	3adant1 1146	. . . . . . . 8 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) ⊆ ℕ)
14		simpr 489	. . . . . . . . . 10 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
15		eqidd 2766	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → (𝑔 ↾ 𝑥) = (𝑔 ↾ 𝑥))
16	8	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → 𝑥 = dom (𝑔 ↾ 𝑥))
17	16	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 = dom (𝑔 ↾ 𝑥))
18		forn 6785	. . . . . . . . . . . 12 ⊢ (𝑔:ℕ–onto→𝐴 → ran 𝑔 = 𝐴)
19	18	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → ran 𝑔 = 𝐴)
20	15, 17, 19	f1oeq123d 6804	. . . . . . . . . 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 235	. . . . . . . . 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 1147	. . . . . . . 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 3461	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
24	23	resex 6018	. . . . . . . . 9 ⊢ (𝑔 ↾ 𝑥) ∈ V
25		dmeq 5883	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → dom 𝑓 = dom (𝑔 ↾ 𝑥))
26	25	sseq1d 3970	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (dom 𝑓 ⊆ ℕ ↔ dom (𝑔 ↾ 𝑥) ⊆ ℕ))
27		id 23	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝑓 = (𝑔 ↾ 𝑥))
28		eqidd 2766	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → 𝐴 = 𝐴)
29	27, 25, 28	f1oeq123d 6804	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴))
30	26, 29	anbi12d 643	. . . . . . . . 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 3568	. . . . . . . 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 595	. . . . . . 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 1135	. . . . . 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 3164	. . . . 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 16	. . . 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 1956	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∃𝑔 𝑔:ℕ–onto→𝐴 → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴)))
38	1, 37	mpd 16	1 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5104   We wwe 5603  dom cdm 5651  ran crn 5652   ↾ cres 5653  –onto→wfo 6523  –1-1-onto→wf1o 6524  ωcom 7850   ≼ cdom 8929   < clt 11231  ℕcn 12221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-n0 12493  df-z 12580  df-uz 12851

This theorem is referenced by:  isomennd  47104