Theorem domomsubct 16012

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 6845	. 2 ⊢ (𝐴 ≼ ω → ∃𝑔 𝑔:𝐴–1-1→ω)
2		imassrn 5038	. . . . 5 ⊢ (𝑔 “ 𝐴) ⊆ ran 𝑔
3		f1rn 5489	. . . . 5 ⊢ (𝑔:𝐴–1-1→ω → ran 𝑔 ⊆ ω)
4	2, 3	sstrid 3205	. . . 4 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 “ 𝐴) ⊆ ω)
5		ssid 3214	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
6		f1ores 5544	. . . . . . . . 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 5490	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1→ω → 𝑔 Fn 𝐴)
9		fnresdm 5390	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝐴 → (𝑔 ↾ 𝐴) = 𝑔)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 ↾ 𝐴) = 𝑔)
11	10	f1oeq1d 5524	. . . . . . . 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 5542	. . . . . . 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 5536	. . . . . 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 2776	. . . . . . 7 ⊢ 𝑔 ∈ V
18	17	cnvex 5226	. . . . . 6 ⊢ ◡𝑔 ∈ V
19		foeq1 5501	. . . . . 6 ⊢ (𝑓 = ◡𝑔 → (𝑓:(𝑔 “ 𝐴)–onto→𝐴 ↔ ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴))
20	18, 19	spcev 2869	. . . . 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 5042	. . . . 5 ⊢ (𝑔 “ 𝐴) ∈ V
23		sseq1 3217	. . . . . 6 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑠 ⊆ ω ↔ (𝑔 “ 𝐴) ⊆ ω))
24		foeq2 5502	. . . . . . 7 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:(𝑔 “ 𝐴)–onto→𝐴))
25	24	exbidv 1849	. . . . . 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 2869	. . . 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 1622	. 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 1373  ∃wex 1516   ⊆ wss 3167   class class class wbr 4047  ωcom 4642  ^◡ccnv 4678  ran crn 4680   ↾ cres 4681   “ cima 4682   Fn wfn 5271  –1-1→wf1 5273  –onto→wfo 5274  –1-1-onto→wf1o 5275   ≼ cdom 6833

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-dom 6836

This theorem is referenced by: (None)