Theorem ssnnf1octb 43536

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 43377	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑔 𝑔:ℕ–onto→𝐴)
2		fofn 6763	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → 𝑔 Fn ℕ)
3		nnex 12168	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → ℕ ∈ V)
5		ltwenn 13877	. . . . . . 7 ⊢ < We ℕ
6	5	a1i 11	. . . . . 6 ⊢ (𝑔:ℕ–onto→𝐴 → < We ℕ)
7	2, 4, 6	wessf1orn 43526	. . . . 5 ⊢ (𝑔:ℕ–onto→𝐴 → ∃𝑥 ∈ 𝒫 ℕ(𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔)
8		f1odm 6793	. . . . . . . . . . 11 ⊢ ((𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔 → dom (𝑔 ↾ 𝑥) = 𝑥)
9	8	adantl 482	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → dom (𝑔 ↾ 𝑥) = 𝑥)
10		elpwi 4572	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 ℕ → 𝑥 ⊆ ℕ)
11	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℕ ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 ⊆ ℕ)
12	9, 11	eqsstrd 3985	. . . . . . . . 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 485	. . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → 𝑥 = dom (𝑔 ↾ 𝑥))
18		forn 6764	. . . . . . . . . . . 12 ⊢ (𝑔:ℕ–onto→𝐴 → ran 𝑔 = 𝐴)
19	18	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔:ℕ–onto→𝐴 ∧ (𝑔 ↾ 𝑥):𝑥–1-1-onto→ran 𝑔) → ran 𝑔 = 𝐴)
20	15, 17, 19	f1oeq123d 6783	. . . . . . . . . 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 231	. . . . . . . . 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 3450	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
24	23	resex 5990	. . . . . . . . 9 ⊢ (𝑔 ↾ 𝑥) ∈ V
25		dmeq 5864	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → dom 𝑓 = dom (𝑔 ↾ 𝑥))
26	25	sseq1d 3978	. . . . . . . . . 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 6783	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ 𝑥) → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ (𝑔 ↾ 𝑥):dom (𝑔 ↾ 𝑥)–1-1-onto→𝐴))
30	26, 29	anbi12d 631	. . . . . . . . 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 3566	. . . . . . . 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 3146	. . . . 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 1936	. 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 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069  Vcvv 3446   ⊆ wss 3913  ∅c0 4287  𝒫 cpw 4565   class class class wbr 5110   We wwe 5592  dom cdm 5638  ran crn 5639   ↾ cres 5640  –onto→wfo 6499  –1-1-onto→wf1o 6500  ωcom 7807   ≼ cdom 8888   < clt 11198  ℕcn 12162

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-n0 12423  df-z 12509  df-uz 12773

This theorem is referenced by:  isomennd  44892