Theorem fo2ndresm 6215

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 2256	. . 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 4689	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5578	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) = ( 2nd ` <. u , v >. ) )
5		vex 2763	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2763	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op2nd 6200	. . . . . . . . . . . 12 |- ( 2nd ` <. u , v >. ) = v
8	4, 7	eqtr2di 2243	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> v = ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) )
9		f2ndres 6213	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
10		ffn 5403	. . . . . . . . . . . . 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 5690	. . . . . . . . . . . 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 2270	. . . . . . . . . 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 3185	. . . . . 6 |- ( E. u u e. A -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5412	. . . . . . 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 3194	. . . . 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 5480	. . 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 2164   C_ wss 3153   <.cop 3621   X. cxp 4657   ran crn 4660   |` cres 4661   Fn wfn 5249   -->wf 5250   -onto->wfo 5252   ` cfv 5254   2ndc2nd 6192

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-2nd 6194

This theorem is referenced by:  2ndconst  6275