Theorem fo2ndres 6714

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 3757	. . . . . . 7 |- ( A =/= (/) <-> E. x x e. A )
2		opelxp 4980	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
3		fvres 5816	. . . . . . . . . . . 12 |- ( <. x , y >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
4		vex 3081	. . . . . . . . . . . . 13 |- x e. _V
5		vex 3081	. . . . . . . . . . . . 13 |- y e. _V
6	4, 5	op2nd 6699	. . . . . . . . . . . 12 |- ( 2nd ` <. x , y >. ) = y
7	3, 6	syl6req 2512	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> y = ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) )
8		f2ndres 6712	. . . . . . . . . . . . 13 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B
9		ffn 5670	. . . . . . . . . . . . 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 5952	. . . . . . . . . . . 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 670	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> ( ( 2nd |` ( A X. B ) ) ` <. x , y >. ) e. ran ( 2nd |` ( A X. B ) ) )
13	7, 12	eqeltrd 2542	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) -> y e. ran ( 2nd |` ( A X. B ) ) )
14	2, 13	sylbir 213	. . . . . . . . 9 |- ( ( x e. A /\ y e. B ) -> y e. ran ( 2nd |` ( A X. B ) ) )
15	14	ex 434	. . . . . . . 8 |- ( x e. A -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
16	15	exlimiv 1689	. . . . . . 7 |- ( E. x x e. A -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
17	1, 16	sylbi 195	. . . . . 6 |- ( A =/= (/) -> ( y e. B -> y e. ran ( 2nd |` ( A X. B ) ) ) )
18	17	ssrdv 3473	. . . . 5 |- ( A =/= (/) -> B C_ ran ( 2nd |` ( A X. B ) ) )
19		frn 5676	. . . . . 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 537	. . . 4 |- ( A =/= (/) -> ( ran ( 2nd |` ( A X. B ) ) C_ B /\ B C_ ran ( 2nd |` ( A X. B ) ) ) )
22		eqss 3482	. . . 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 212	. . 3 |- ( A =/= (/) -> ran ( 2nd |` ( A X. B ) ) = B )
24	23, 8	jctil 537	. 2 |- ( A =/= (/) -> ( ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B /\ ran ( 2nd |` ( A X. B ) ) = B ) )
25		dffo2 5735	. 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 212	1 |- ( A =/= (/) -> ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2648   C_ wss 3439   (/)c0 3748   <.cop 3994   X. cxp 4949   ran crn 4952   |` cres 4953   Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529   2ndc2nd 6689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-2nd 6691

This theorem is referenced by:  2ndconst  6775  txcmpb  19359