Theorem fo1stres 7192

Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
fo1stres	|- ( B =/= (/) -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Proof of Theorem fo1stres

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

Step	Hyp	Ref	Expression
1		n0 3931	. . . . . . 7 |- ( B =/= (/) <-> E. y y e. B )
2		opelxp 5146	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
3		fvres 6207	. . . . . . . . . . . 12 |- ( <. x , y >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. x , y >. ) = ( 1st ` <. x , y >. ) )
4		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
5		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
6	4, 5	op1st 7176	. . . . . . . . . . . 12 |- ( 1st ` <. x , y >. ) = x
7	3, 6	syl6req 2673	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> x = ( ( 1st |` ( A X. B ) ) ` <. x , y >. ) )
8		f1stres 7190	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
9		ffn 6045	. . . . . . . . . . . . 13 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ( 1st |` ( A X. B ) ) Fn ( A X. B ) )
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 |- ( 1st |` ( A X. B ) ) Fn ( A X. B )
11		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( ( 1st |` ( A X. B ) ) Fn ( A X. B ) /\ <. x , y >. e. ( A X. B ) ) -> ( ( 1st |` ( A X. B ) ) ` <. x , y >. ) e. ran ( 1st |` ( A X. B ) ) )
12	10, 11	mpan 706	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. x , y >. ) e. ran ( 1st |` ( A X. B ) ) )
13	7, 12	eqeltrd 2701	. . . . . . . . . 10 |- ( <. x , y >. e. ( A X. B ) -> x e. ran ( 1st |` ( A X. B ) ) )
14	2, 13	sylbir 225	. . . . . . . . 9 |- ( ( x e. A /\ y e. B ) -> x e. ran ( 1st |` ( A X. B ) ) )
15	14	expcom 451	. . . . . . . 8 |- ( y e. B -> ( x e. A -> x e. ran ( 1st |` ( A X. B ) ) ) )
16	15	exlimiv 1858	. . . . . . 7 |- ( E. y y e. B -> ( x e. A -> x e. ran ( 1st |` ( A X. B ) ) ) )
17	1, 16	sylbi 207	. . . . . 6 |- ( B =/= (/) -> ( x e. A -> x e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 3609	. . . . 5 |- ( B =/= (/) -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 6053	. . . . . 6 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ran ( 1st |` ( A X. B ) ) C_ A )
20	8, 19	ax-mp 5	. . . . 5 |- ran ( 1st |` ( A X. B ) ) C_ A
21	18, 20	jctil 560	. . . 4 |- ( B =/= (/) -> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
22		eqss 3618	. . . 4 |- ( ran ( 1st |` ( A X. B ) ) = A <-> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
23	21, 22	sylibr 224	. . 3 |- ( B =/= (/) -> ran ( 1st |` ( A X. B ) ) = A )
24	23, 8	jctil 560	. 2 |- ( B =/= (/) -> ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A /\ ran ( 1st |` ( A X. B ) ) = A ) )
25		dffo2 6119	. 2 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A <-> ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A /\ ran ( 1st |` ( A X. B ) ) = A ) )
26	24, 25	sylibr 224	1 |- ( B =/= (/) -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   <.cop 4183   X. cxp 5112   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168

This theorem is referenced by:  1stconst  7265  txcmpb  21447