Theorem fo1stresm 5788

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 2100	. . 3 |- ( v = y -> ( v e. B <-> y e. B ) )
2	1	cbvexv 1795	. 2 |- ( E. v v e. B <-> E. y y e. B )
3		opelxp 4374	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5198	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) = ( 1st ` <. u , v >. ) )
5		vex 2560	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2560	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op1st 5773	. . . . . . . . . . . 12 |- ( 1st ` <. u , v >. ) = u
8	4, 7	syl6req 2089	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> u = ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) )
9		f1stres 5786	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
10		ffn 5046	. . . . . . . . . . . . 13 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ( 1st |` ( A X. B ) ) Fn ( A X. B ) )
11	9, 10	ax-mp 7	. . . . . . . . . . . 12 |- ( 1st |` ( A X. B ) ) Fn ( A X. B )
12		fnfvelrn 5299	. . . . . . . . . . . 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 400	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 1st |` ( A X. B ) ) )
14	8, 13	eqeltrd 2114	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
15	3, 14	sylbir 125	. . . . . . . . 9 |- ( ( u e. A /\ v e. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
16	15	expcom 109	. . . . . . . 8 |- ( v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
17	16	exlimiv 1489	. . . . . . 7 |- ( E. v v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 2951	. . . . . 6 |- ( E. v v e. B -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 5052	. . . . . . 7 |- ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A -> ran ( 1st |` ( A X. B ) ) C_ A )
20	9, 19	ax-mp 7	. . . . . 6 |- ran ( 1st |` ( A X. B ) ) C_ A
21	18, 20	jctil 295	. . . . 5 |- ( E. v v e. B -> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
22		eqss 2960	. . . . 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 137	. . . 4 |- ( E. v v e. B -> ran ( 1st |` ( A X. B ) ) = A )
24	23, 9	jctil 295	. . 3 |- ( E. v v e. B -> ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A /\ ran ( 1st |` ( A X. B ) ) = A ) )
25		dffo2 5110	. . 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 137	. 2 |- ( E. v v e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )
27	2, 26	sylbir 125	1 |- ( E. y y e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   X. cxp 4343   ran crn 4346   |` cres 4347   Fn wfn 4897   -->wf 4898   -onto->wfo 4900   ` cfv 4902   1stc1st 5765

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767

This theorem is referenced by:  1stconst  5842