Theorem domomsubct 16278

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

Assertion

Ref	Expression
domomsubct	|- ( A ~<_ om -> E. s ( s C_ om /\ E. 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 ~<_ om -> E. g g : A -1-1-> om )
2		imassrn 5055	. . . . 5 |- ( g " A ) C_ ran g
3		f1rn 5508	. . . . 5 |- ( g : A -1-1-> om -> ran g C_ om )
4	2, 3	sstrid 3215	. . . 4 |- ( g : A -1-1-> om -> ( g " A ) C_ om )
5		ssid 3224	. . . . . . . . 9 |- A C_ A
6		f1ores 5563	. . . . . . . . 9 |- ( ( g : A -1-1-> om /\ A C_ A ) -> ( g |` A ) : A -1-1-onto-> ( g " A ) )
7	5, 6	mpan2 425	. . . . . . . 8 |- ( g : A -1-1-> om -> ( g |` A ) : A -1-1-onto-> ( g " A ) )
8		f1fn 5509	. . . . . . . . . 10 |- ( g : A -1-1-> om -> g Fn A )
9		fnresdm 5408	. . . . . . . . . 10 |- ( g Fn A -> ( g |` A ) = g )
10	8, 9	syl 14	. . . . . . . . 9 |- ( g : A -1-1-> om -> ( g |` A ) = g )
11	10	f1oeq1d 5543	. . . . . . . 8 |- ( g : A -1-1-> om -> ( ( 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-> om -> 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-> om -> `' 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-> om -> `' g : ( g " A ) -onto-> A )
17		vex 2782	. . . . . . 7 |- g e. _V
18	17	cnvex 5243	. . . . . 6 |- `' g e. _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 -> E. f f : ( g " A ) -onto-> A )
21	16, 20	syl 14	. . . 4 |- ( g : A -1-1-> om -> E. f f : ( g " A ) -onto-> A )
22	17	imaex 5059	. . . . 5 |- ( g " A ) e. _V
23		sseq1 3227	. . . . . 6 |- ( s = ( g " A ) -> ( s C_ om <-> ( g " A ) C_ om ) )
24		foeq2 5521	. . . . . . 7 |- ( s = ( g " A ) -> ( f : s -onto-> A <-> f : ( g " A ) -onto-> A ) )
25	24	exbidv 1851	. . . . . 6 |- ( s = ( g " A ) -> ( E. f f : s -onto-> A <-> E. f f : ( g " A ) -onto-> A ) )
26	23, 25	anbi12d 473	. . . . 5 |- ( s = ( g " A ) -> ( ( s C_ om /\ E. f f : s -onto-> A ) <-> ( ( g " A ) C_ om /\ E. f f : ( g " A ) -onto-> A ) ) )
27	22, 26	spcev 2878	. . . 4 |- ( ( ( g " A ) C_ om /\ E. f f : ( g " A ) -onto-> A ) -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
28	4, 21, 27	syl2anc 411	. . 3 |- ( g : A -1-1-> om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
29	28	exlimiv 1624	. 2 |- ( E. g g : A -1-1-> om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
30	1, 29	syl 14	1 |- ( A ~<_ om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1375   E.wex 1518   C_ wss 3177   class class class wbr 4062   omcom 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)