Theorem fo2ndresm 6181

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 2252	. . 3 |- ( u = x -> ( u e. A <-> x e. A ) )
2	1	cbvexv 1930	. 2 |- ( E. u u e. A <-> E. x x e. A )
3		opelxp 4671	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5554	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) = ( 2nd ` <. u , v >. ) )
5		vex 2755	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2755	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op2nd 6166	. . . . . . . . . . . 12 |- ( 2nd ` <. u , v >. ) = v
8	4, 7	eqtr2di 2239	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> v = ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) )
9		f2ndres 6179	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
10		ffn 5380	. . . . . . . . . . . . 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 5664	. . . . . . . . . . . 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 2266	. . . . . . . . . 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 1609	. . . . . . 7 |- ( E. u u e. A -> ( v e. B -> v e. ran ( 2nd |` ( A X. B ) ) ) )
18	17	ssrdv 3176	. . . . . 6 |- ( E. u u e. A -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5389	. . . . . . 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 3185	. . . . 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 5457	. . 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 1364   E.wex 1503   e. wcel 2160   C_ wss 3144   <.cop 3610   X. cxp 4639   ran crn 4642   |` cres 4643   Fn wfn 5226   -->wf 5227   -onto->wfo 5229   ` cfv 5231   2ndc2nd 6158

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239  df-2nd 6160

This theorem is referenced by:  2ndconst  6241