Theorem domomsubct 16916

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 7000	. 2 |- ( A ~<_ om -> E. g g : A -1-1-> om )
2		imassrn 5118	. . . . 5 |- ( g " A ) C_ ran g
3		f1rn 5580	. . . . 5 |- ( g : A -1-1-> om -> ran g C_ om )
4	2, 3	sstrid 3253	. . . 4 |- ( g : A -1-1-> om -> ( g " A ) C_ om )
5		ssid 3262	. . . . . . . . 9 |- A C_ A
6		f1ores 5635	. . . . . . . . 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 5581	. . . . . . . . . 10 |- ( g : A -1-1-> om -> g Fn A )
9		fnresdm 5473	. . . . . . . . . 10 |- ( g Fn A -> ( g |` A ) = g )
10	8, 9	syl 14	. . . . . . . . 9 |- ( g : A -1-1-> om -> ( g |` A ) = g )
11	10	f1oeq1d 5615	. . . . . . . 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 5633	. . . . . . 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 5627	. . . . . 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 2818	. . . . . . 7 |- g e. _V
18	17	cnvex 5307	. . . . . 6 |- `' g e. _V
19		foeq1 5592	. . . . . 6 |- ( f = `' g -> ( f : ( g " A ) -onto-> A <-> `' g : ( g " A ) -onto-> A ) )
20	18, 19	spcev 2914	. . . . 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 5122	. . . . 5 |- ( g " A ) e. _V
23		sseq1 3265	. . . . . 6 |- ( s = ( g " A ) -> ( s C_ om <-> ( g " A ) C_ om ) )
24		foeq2 5593	. . . . . . 7 |- ( s = ( g " A ) -> ( f : s -onto-> A <-> f : ( g " A ) -onto-> A ) )
25	24	exbidv 1874	. . . . . 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 2914	. . . 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 1647	. 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 1398   E.wex 1541   C_ wss 3214   class class class wbr 4115   omcom 4718   `'ccnv 4754   ran crn 4756   |` cres 4757   "cima 4758   Fn wfn 5353   -1-1->wf1 5355   -onto->wfo 5356   -1-1-onto->wf1o 5357   ~<_ cdom 6988

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-dom 6991

This theorem is referenced by: (None)