Theorem fo2ndres 6818

Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
fo2ndres	|- ( A =/= (/) -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )

Proof of Theorem fo2ndres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3741	. . . . . . 7 |- ( A =/= (/) <-> E. x x e. A )
2		opelxp 4864	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
3		fvres 5879	. . . . . . . . . . . 12 |- ( <. x , y >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
4		vex 3048	. . . . . . . . . . . . 13 |- x e. _V
5		vex 3048	. . . . . . . . . . . . 13 |- y e. _V
6	4, 5	op2nd 6802	. . . . . . . . . . . 12 |- ( 2nd ` <. x , y >. ) = y
7	3, 6	syl6req 2502	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> y = ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) )
8		f2ndres 6816	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
9		ffn 5728	. . . . . . . . . . . . 13 |- ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B -> ( 2nd |` ( A X. B ) ) Fn ( A X. B ) )
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 |- ( 2nd |` ( A X. B ) ) Fn ( A X. B )
11		fnfvelrn 6019	. . . . . . . . . . . 12 |- ( ( ( 2nd |` ( A X. B ) ) Fn ( A X. B ) /\ <. x , y >. e. ( A X. B ) ) -> ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) e. ran ( 2nd |` ( A X. B ) ) )
12	10, 11	mpan 676	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) e. ran ( 2nd |` ( A X. B ) ) )
13	7, 12	eqeltrd 2529	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) -> y e. ran ( 2nd |` ( A X. B ) ) )
14	2, 13	sylbir 217	. . . . . . . . 9 |- ( ( x e. A /\ y e. B ) -> y e. ran ( 2nd |` ( A X. B ) ) )
15	14	ex 436	. . . . . . . 8 |- ( x e. A -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
16	15	exlimiv 1776	. . . . . . 7 |- ( E. x x e. A -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
17	1, 16	sylbi 199	. . . . . 6 |- ( A =/= (/) -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
18	17	ssrdv 3438	. . . . 5 |- ( A =/= (/) -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5735	. . . . . 6 |- ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B -> ran ( 2nd |` ( A X. B ) ) C_ B )
20	8, 19	ax-mp 5	. . . . 5 |- ran ( 2nd |` ( A X. B ) ) C_ B
21	18, 20	jctil 540	. . . 4 |- ( A =/= (/) -> ( ran ( 2nd |` ( A X. B ) ) C_ B /\ B C_ ran ( 2nd |` ( A X. B ) ) ) )
22		eqss 3447	. . . 4 |- ( ran ( 2nd |` ( A X. B ) ) = B <-> ( ran ( 2nd |` ( A X. B ) ) C_ B /\ B C_ ran ( 2nd |` ( A X. B ) ) ) )
23	21, 22	sylibr 216	. . 3 |- ( A =/= (/) -> ran ( 2nd |` ( A X. B ) ) = B )
24	23, 8	jctil 540	. 2 |- ( A =/= (/) -> ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B /\ ran ( 2nd |` ( A X. B ) ) = B ) )
25		dffo2 5797	. 2 |- ( ( 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 216	1 |- ( A =/= (/) -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   C_ wss 3404   (/)c0 3731   <.cop 3974   X. cxp 4832   ran crn 4835   |` cres 4836   Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582   2ndc2nd 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-2nd 6794

This theorem is referenced by:  2ndconst  6885  txcmpb  20659