Theorem domomsubct 17014

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 7027	. 2 |- ( A ~<_ om -> E. g g : A -1-1-> om )
2		imassrn 5135	. . . . 5 |- ( g " A ) C_ ran g
3		f1rn 5597	. . . . 5 |- ( g : A -1-1-> om -> ran g C_ om )
4	2, 3	sstrid 3259	. . . 4 |- ( g : A -1-1-> om -> ( g " A ) C_ om )
5		ssid 3268	. . . . . . . . 9 |- A C_ A
6		f1ores 5652	. . . . . . . . 9 |- ( ( g : A -1-1-> om /\ A C_ A ) -> ( g |` A ) : A -1-1-onto-> ( g " A ) )
7	5, 6	mpan2 429	. . . . . . . 8 |- ( g : A -1-1-> om -> ( g |` A ) : A -1-1-onto-> ( g " A ) )
8		f1fn 5598	. . . . . . . . . 10 |- ( g : A -1-1-> om -> g Fn A )
9		fnresdm 5490	. . . . . . . . . 10 |- ( g Fn A -> ( g |` A ) = g )
10	8, 9	syl 14	. . . . . . . . 9 |- ( g : A -1-1-> om -> ( g |` A ) = g )
11	10	f1oeq1d 5632	. . . . . . . 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 5650	. . . . . . 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 5644	. . . . . 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 2824	. . . . . . 7 |- g e. _V
18	17	cnvex 5324	. . . . . 6 |- `' g e. _V
19		foeq1 5609	. . . . . 6 |- ( f = `' g -> ( f : ( g " A ) -onto-> A <-> `' g : ( g " A ) -onto-> A ) )
20	18, 19	spcev 2920	. . . . 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 5139	. . . . 5 |- ( g " A ) e. _V
23		sseq1 3271	. . . . . 6 |- ( s = ( g " A ) -> ( s C_ om <-> ( g " A ) C_ om ) )
24		foeq2 5610	. . . . . . 7 |- ( s = ( g " A ) -> ( f : s -onto-> A <-> f : ( g " A ) -onto-> A ) )
25	24	exbidv 1878	. . . . . 6 |- ( s = ( g " A ) -> ( E. f f : s -onto-> A <-> E. f f : ( g " A ) -onto-> A ) )
26	23, 25	anbi12d 477	. . . . 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 2920	. . . 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 415	. . 3 |- ( g : A -1-1-> om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
29	28	exlimiv 1651	. 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 1402   E.wex 1545   C_ wss 3220   class class class wbr 4128   omcom 4735   `'ccnv 4771   ran crn 4773   |` cres 4774   "cima 4775   Fn wfn 5370   -1-1->wf1 5372   -onto->wfo 5373   -1-1-onto->wf1o 5374   ~<_ cdom 7015

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-dom 7018

This theorem is referenced by: (None)