Theorem domomsubct 16662

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 6925	. 2 ⊢ (𝐴 ≼ ω → ∃𝑔 𝑔:𝐴–1-1→ω)
2		imassrn 5089	. . . . 5 ⊢ (𝑔 “ 𝐴) ⊆ ran 𝑔
3		f1rn 5546	. . . . 5 ⊢ (𝑔:𝐴–1-1→ω → ran 𝑔 ⊆ ω)
4	2, 3	sstrid 3237	. . . 4 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 “ 𝐴) ⊆ ω)
5		ssid 3246	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
6		f1ores 5601	. . . . . . . . 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 5547	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1→ω → 𝑔 Fn 𝐴)
9		fnresdm 5443	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝐴 → (𝑔 ↾ 𝐴) = 𝑔)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 ↾ 𝐴) = 𝑔)
11	10	f1oeq1d 5581	. . . . . . . 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 5599	. . . . . . 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 5593	. . . . . 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 2804	. . . . . . 7 ⊢ 𝑔 ∈ V
18	17	cnvex 5277	. . . . . 6 ⊢ ◡𝑔 ∈ V
19		foeq1 5558	. . . . . 6 ⊢ (𝑓 = ◡𝑔 → (𝑓:(𝑔 “ 𝐴)–onto→𝐴 ↔ ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴))
20	18, 19	spcev 2900	. . . . 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 5093	. . . . 5 ⊢ (𝑔 “ 𝐴) ∈ V
23		sseq1 3249	. . . . . 6 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑠 ⊆ ω ↔ (𝑔 “ 𝐴) ⊆ ω))
24		foeq2 5559	. . . . . . 7 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:(𝑔 “ 𝐴)–onto→𝐴))
25	24	exbidv 1872	. . . . . 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 2900	. . . 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 1646	. 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 1397  ∃wex 1540   ⊆ wss 3199   class class class wbr 4089  ωcom 4690  ^◡ccnv 4726  ran crn 4728   ↾ cres 4729   “ cima 4730   Fn wfn 5323  –1-1→wf1 5325  –onto→wfo 5326  –1-1-onto→wf1o 5327   ≼ cdom 6913

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-dom 6916

This theorem is referenced by: (None)