Theorem fo2ndresm 6386

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 2301	. . 3 |- ( u = x -> ( u e. A <-> x e. A ) )
2	1	cbvexv 1974	. 2 |- ( E. u u e. A <-> E. x x e. A )
3		opelxp 4799	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5714	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) = ( 2nd ` <. u , v >. ) )
5		vex 2824	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2824	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op2nd 6371	. . . . . . . . . . . 12 |- ( 2nd ` <. u , v >. ) = v
8	4, 7	eqtr2di 2288	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> v = ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) )
9		f2ndres 6384	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
10		ffn 5528	. . . . . . . . . . . . 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 5831	. . . . . . . . . . . 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 428	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 2nd |` ( A X. B ) ) )
14	8, 13	eqeltrd 2315	. . . . . . . . . 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 1651	. . . . . . 7 |- ( E. u u e. A -> ( v e. B -> v e. ran ( 2nd |` ( A X. B ) ) ) )
18	17	ssrdv 3254	. . . . . 6 |- ( E. u u e. A -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5537	. . . . . . 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 3263	. . . . 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 5614	. . 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 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3708   X. cxp 4767   ran crn 4770   |` cres 4771   Fn wfn 5367   -->wf 5368   -onto->wfo 5370   ` cfv 5372   2ndc2nd 6363

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-2nd 6365

This theorem is referenced by:  2ndconst  6448