Theorem domomsubct 16278

Description: A set dominated by ω is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)

Assertion

Ref	Expression
domomsubct	⊢ (𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))

Distinct variable group:   𝐴,𝑓,𝑠

Proof of Theorem domomsubct

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6868	. 2 ⊢ (𝐴 ≼ ω → ∃𝑔 𝑔:𝐴–1-1→ω)
2		imassrn 5055	. . . . 5 ⊢ (𝑔 “ 𝐴) ⊆ ran 𝑔
3		f1rn 5508	. . . . 5 ⊢ (𝑔:𝐴–1-1→ω → ran 𝑔 ⊆ ω)
4	2, 3	sstrid 3215	. . . 4 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 “ 𝐴) ⊆ ω)
5		ssid 3224	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
6		f1ores 5563	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1→ω ∧ 𝐴 ⊆ 𝐴) → (𝑔 ↾ 𝐴):𝐴–1-1-onto→(𝑔 “ 𝐴))
7	5, 6	mpan2 425	. . . . . . . 8 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 ↾ 𝐴):𝐴–1-1-onto→(𝑔 “ 𝐴))
8		f1fn 5509	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1→ω → 𝑔 Fn 𝐴)
9		fnresdm 5408	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝐴 → (𝑔 ↾ 𝐴) = 𝑔)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 ↾ 𝐴) = 𝑔)
11	10	f1oeq1d 5543	. . . . . . . 8 ⊢ (𝑔:𝐴–1-1→ω → ((𝑔 ↾ 𝐴):𝐴–1-1-onto→(𝑔 “ 𝐴) ↔ 𝑔:𝐴–1-1-onto→(𝑔 “ 𝐴)))
12	7, 11	mpbid 147	. . . . . . 7 ⊢ (𝑔:𝐴–1-1→ω → 𝑔:𝐴–1-1-onto→(𝑔 “ 𝐴))
13		f1ocnv 5561	. . . . . . 7 ⊢ (𝑔:𝐴–1-1-onto→(𝑔 “ 𝐴) → ◡𝑔:(𝑔 “ 𝐴)–1-1-onto→𝐴)
14	12, 13	syl 14	. . . . . 6 ⊢ (𝑔:𝐴–1-1→ω → ◡𝑔:(𝑔 “ 𝐴)–1-1-onto→𝐴)
15		f1ofo 5555	. . . . . 6 ⊢ (◡𝑔:(𝑔 “ 𝐴)–1-1-onto→𝐴 → ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴)
16	14, 15	syl 14	. . . . 5 ⊢ (𝑔:𝐴–1-1→ω → ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴)
17		vex 2782	. . . . . . 7 ⊢ 𝑔 ∈ V
18	17	cnvex 5243	. . . . . 6 ⊢ ◡𝑔 ∈ V
19		foeq1 5520	. . . . . 6 ⊢ (𝑓 = ◡𝑔 → (𝑓:(𝑔 “ 𝐴)–onto→𝐴 ↔ ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴))
20	18, 19	spcev 2878	. . . . 5 ⊢ (◡𝑔:(𝑔 “ 𝐴)–onto→𝐴 → ∃𝑓 𝑓:(𝑔 “ 𝐴)–onto→𝐴)
21	16, 20	syl 14	. . . 4 ⊢ (𝑔:𝐴–1-1→ω → ∃𝑓 𝑓:(𝑔 “ 𝐴)–onto→𝐴)
22	17	imaex 5059	. . . . 5 ⊢ (𝑔 “ 𝐴) ∈ V
23		sseq1 3227	. . . . . 6 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑠 ⊆ ω ↔ (𝑔 “ 𝐴) ⊆ ω))
24		foeq2 5521	. . . . . . 7 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:(𝑔 “ 𝐴)–onto→𝐴))
25	24	exbidv 1851	. . . . . 6 ⊢ (𝑠 = (𝑔 “ 𝐴) → (∃𝑓 𝑓:𝑠–onto→𝐴 ↔ ∃𝑓 𝑓:(𝑔 “ 𝐴)–onto→𝐴))
26	23, 25	anbi12d 473	. . . . 5 ⊢ (𝑠 = (𝑔 “ 𝐴) → ((𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴) ↔ ((𝑔 “ 𝐴) ⊆ ω ∧ ∃𝑓 𝑓:(𝑔 “ 𝐴)–onto→𝐴)))
27	22, 26	spcev 2878	. . . 4 ⊢ (((𝑔 “ 𝐴) ⊆ ω ∧ ∃𝑓 𝑓:(𝑔 “ 𝐴)–onto→𝐴) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
28	4, 21, 27	syl2anc 411	. . 3 ⊢ (𝑔:𝐴–1-1→ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
29	28	exlimiv 1624	. 2 ⊢ (∃𝑔 𝑔:𝐴–1-1→ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
30	1, 29	syl 14	1 ⊢ (𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375  ∃wex 1518   ⊆ wss 3177   class class class wbr 4062  ωcom 4659  ^◡ccnv 4695  ran crn 4697   ↾ cres 4698   “ cima 4699   Fn wfn 5289  –1-1→wf1 5291  –onto→wfo 5292  –1-1-onto→wf1o 5293   ≼ cdom 6856

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-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-dom 6859

This theorem is referenced by: (None)