Theorem domomsubct 16623

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 6920	. 2 |- ( A ~<_ om -> E. g g : A -1-1-> om )
2		imassrn 5087	. . . . 5 |- ( g " A ) C_ ran g
3		f1rn 5543	. . . . 5 |- ( g : A -1-1-> om -> ran g C_ om )
4	2, 3	sstrid 3238	. . . 4 |- ( g : A -1-1-> om -> ( g " A ) C_ om )
5		ssid 3247	. . . . . . . . 9 |- A C_ A
6		f1ores 5598	. . . . . . . . 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 5544	. . . . . . . . . 10 |- ( g : A -1-1-> om -> g Fn A )
9		fnresdm 5441	. . . . . . . . . 10 |- ( g Fn A -> ( g |` A ) = g )
10	8, 9	syl 14	. . . . . . . . 9 |- ( g : A -1-1-> om -> ( g |` A ) = g )
11	10	f1oeq1d 5578	. . . . . . . 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 5596	. . . . . . 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 5590	. . . . . 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 2805	. . . . . . 7 |- g e. _V
18	17	cnvex 5275	. . . . . 6 |- `' g e. _V
19		foeq1 5555	. . . . . 6 |- ( f = `' g -> ( f : ( g " A ) -onto-> A <-> `' g : ( g " A ) -onto-> A ) )
20	18, 19	spcev 2901	. . . . 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 5091	. . . . 5 |- ( g " A ) e. _V
23		sseq1 3250	. . . . . 6 |- ( s = ( g " A ) -> ( s C_ om <-> ( g " A ) C_ om ) )
24		foeq2 5556	. . . . . . 7 |- ( s = ( g " A ) -> ( f : s -onto-> A <-> f : ( g " A ) -onto-> A ) )
25	24	exbidv 1873	. . . . . 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 2901	. . . 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 1646	. 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 1397   E.wex 1540   C_ wss 3200   class class class wbr 4088   omcom 4688   `'ccnv 4724   ran crn 4726   |` cres 4727   "cima 4728   Fn wfn 5321   -1-1->wf1 5323   -onto->wfo 5324   -1-1-onto->wf1o 5325   ~<_ cdom 6908

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-dom 6911

This theorem is referenced by: (None)