Theorem fo1stresm 6323

Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Assertion

Ref	Expression
fo1stresm	|- ( E. y y e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Distinct variable group:   y, B

Allowed substitution hint:   A( y)

Proof of Theorem fo1stresm

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2294	. . 3 |- ( v = y -> ( v e. B <-> y e. B ) )
2	1	cbvexv 1967	. 2 |- ( E. v v e. B <-> E. y y e. B )
3		opelxp 4755	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5663	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) = ( 1st ` <. u , v >. ) )
5		vex 2805	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2805	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op1st 6308	. . . . . . . . . . . 12 |- ( 1st ` <. u , v >. ) = u
8	4, 7	eqtr2di 2281	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> u = ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) )
9		f1stres 6321	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
10		ffn 5482	. . . . . . . . . . . . 13 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ( 1st |` ( A X. B ) ) Fn ( A X. B ) )
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 |- ( 1st |` ( A X. B ) ) Fn ( A X. B )
12		fnfvelrn 5779	. . . . . . . . . . . 12 |- ( ( ( 1st |` ( A X. B ) ) Fn ( A X. B ) /\ <. u , v >. e. ( A X. B ) ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 1st |` ( A X. B ) ) )
13	11, 12	mpan 424	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 1st |` ( A X. B ) ) )
14	8, 13	eqeltrd 2308	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
15	3, 14	sylbir 135	. . . . . . . . 9 |- ( ( u e. A /\ v e. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
16	15	expcom 116	. . . . . . . 8 |- ( v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
17	16	exlimiv 1646	. . . . . . 7 |- ( E. v v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 3233	. . . . . 6 |- ( E. v v e. B -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 5491	. . . . . . 7 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ran ( 1st |` ( A X. B ) ) C_ A )
20	9, 19	ax-mp 5	. . . . . 6 |- ran ( 1st |` ( A X. B ) ) C_ A
21	18, 20	jctil 312	. . . . 5 |- ( E. v v e. B -> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
22		eqss 3242	. . . . 5 |- ( ran ( 1st |` ( A X. B ) ) = A <-> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
23	21, 22	sylibr 134	. . . 4 |- ( E. v v e. B -> ran ( 1st |` ( A X. B ) ) = A )
24	23, 9	jctil 312	. . 3 |- ( E. v v e. B -> ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A /\ ran ( 1st |` ( A X. B ) ) = A ) )
25		dffo2 5563	. . 3 |- ( ( 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 134	. 2 |- ( E. v v e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )
27	2, 26	sylbir 135	1 |- ( E. y y e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   C_ wss 3200   <.cop 3672   X. cxp 4723   ran crn 4726   |` cres 4727   Fn wfn 5321   -->wf 5322   -onto->wfo 5324   ` cfv 5326   1stc1st 6300

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302

This theorem is referenced by:  1stconst  6385