Theorem fo2ndresm 6308

Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Assertion

Ref	Expression
fo2ndresm	|- ( E. x x e. A -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem fo2ndresm

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2292	. . 3 |- ( u = x -> ( u e. A <-> x e. A ) )
2	1	cbvexv 1965	. 2 |- ( E. u u e. A <-> E. x x e. A )
3		opelxp 4749	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5651	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) = ( 2nd ` <. u , v >. ) )
5		vex 2802	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2802	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op2nd 6293	. . . . . . . . . . . 12 |- ( 2nd ` <. u , v >. ) = v
8	4, 7	eqtr2di 2279	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> v = ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) )
9		f2ndres 6306	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
10		ffn 5473	. . . . . . . . . . . . 13 |- ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B -> ( 2nd |` ( A X. B ) ) Fn ( A X. B ) )
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 |- ( 2nd |` ( A X. B ) ) Fn ( A X. B )
12		fnfvelrn 5767	. . . . . . . . . . . 12 |- ( ( ( 2nd |` ( A X. B ) ) Fn ( A X. B ) /\ <. u , v >. e. ( A X. B ) ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 2nd |` ( A X. B ) ) )
13	11, 12	mpan 424	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 2nd |` ( A X. B ) ) )
14	8, 13	eqeltrd 2306	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) -> v e. ran ( 2nd |` ( A X. B ) ) )
15	3, 14	sylbir 135	. . . . . . . . 9 |- ( ( u e. A /\ v e. B ) -> v e. ran ( 2nd |` ( A X. B ) ) )
16	15	ex 115	. . . . . . . 8 |- ( u e. A -> ( v e. B -> v e. ran ( 2nd |` ( A X. B ) ) ) )
17	16	exlimiv 1644	. . . . . . 7 |- ( E. u u e. A -> ( v e. B -> v e. ran ( 2nd |` ( A X. B ) ) ) )
18	17	ssrdv 3230	. . . . . 6 |- ( E. u u e. A -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5482	. . . . . . 7 |- ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B -> ran ( 2nd |` ( A X. B ) ) C_ B )
20	9, 19	ax-mp 5	. . . . . 6 |- ran ( 2nd |` ( A X. B ) ) C_ B
21	18, 20	jctil 312	. . . . 5 |- ( E. u u e. A -> ( ran ( 2nd |` ( A X. B ) ) C_ B /\ B C_ ran ( 2nd |` ( A X. B ) ) ) )
22		eqss 3239	. . . . 5 |- ( ran ( 2nd |` ( A X. B ) ) = B <-> ( ran ( 2nd |` ( A X. B ) ) C_ B /\ B C_ ran ( 2nd |` ( A X. B ) ) ) )
23	21, 22	sylibr 134	. . . 4 |- ( E. u u e. A -> ran ( 2nd |` ( A X. B ) ) = B )
24	23, 9	jctil 312	. . 3 |- ( E. u u e. A -> ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B /\ ran ( 2nd |` ( A X. B ) ) = B ) )
25		dffo2 5552	. . 3 |- ( ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B <-> ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B /\ ran ( 2nd |` ( A X. B ) ) = B ) )
26	24, 25	sylibr 134	. 2 |- ( E. u u e. A -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )
27	2, 26	sylbir 135	1 |- ( E. x x e. A -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3197   <.cop 3669   X. cxp 4717   ran crn 4720   |` cres 4721   Fn wfn 5313   -->wf 5314   -onto->wfo 5316   ` cfv 5318   2ndc2nd 6285

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-2nd 6287

This theorem is referenced by:  2ndconst  6368