Theorem domomsubct 15732

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 6817	. 2 ⊢ (𝐴 ≼ ω → ∃𝑔 𝑔:𝐴–1-1→ω)
2		imassrn 5021	. . . . 5 ⊢ (𝑔 “ 𝐴) ⊆ ran 𝑔
3		f1rn 5467	. . . . 5 ⊢ (𝑔:𝐴–1-1→ω → ran 𝑔 ⊆ ω)
4	2, 3	sstrid 3195	. . . 4 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 “ 𝐴) ⊆ ω)
5		ssid 3204	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
6		f1ores 5522	. . . . . . . . 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 5468	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1→ω → 𝑔 Fn 𝐴)
9		fnresdm 5370	. . . . . . . . . 10 ⊢ (𝑔 Fn 𝐴 → (𝑔 ↾ 𝐴) = 𝑔)
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1→ω → (𝑔 ↾ 𝐴) = 𝑔)
11	10	f1oeq1d 5502	. . . . . . . 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 5520	. . . . . . 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 5514	. . . . . 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 2766	. . . . . . 7 ⊢ 𝑔 ∈ V
18	17	cnvex 5209	. . . . . 6 ⊢ ◡𝑔 ∈ V
19		foeq1 5479	. . . . . 6 ⊢ (𝑓 = ◡𝑔 → (𝑓:(𝑔 “ 𝐴)–onto→𝐴 ↔ ◡𝑔:(𝑔 “ 𝐴)–onto→𝐴))
20	18, 19	spcev 2859	. . . . 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 5025	. . . . 5 ⊢ (𝑔 “ 𝐴) ∈ V
23		sseq1 3207	. . . . . 6 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑠 ⊆ ω ↔ (𝑔 “ 𝐴) ⊆ ω))
24		foeq2 5480	. . . . . . 7 ⊢ (𝑠 = (𝑔 “ 𝐴) → (𝑓:𝑠–onto→𝐴 ↔ 𝑓:(𝑔 “ 𝐴)–onto→𝐴))
25	24	exbidv 1839	. . . . . 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 2859	. . . 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 1612	. 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 1364  ∃wex 1506   ⊆ wss 3157   class class class wbr 4034  ωcom 4627  ^◡ccnv 4663  ran crn 4665   ↾ cres 4666   “ cima 4667   Fn wfn 5254  –1-1→wf1 5256  –onto→wfo 5257  –1-1-onto→wf1o 5258   ≼ cdom 6807

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-dom 6810

This theorem is referenced by: (None)