Theorem fo1stresm 6333

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 4761	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5672	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) = ( 1st ` <. u , v >. ) )
5		vex 2806	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2806	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op1st 6318	. . . . . . . . . . . 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 6331	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
10		ffn 5489	. . . . . . . . . . . . 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 5787	. . . . . . . . . . . 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 1647	. . . . . . 7 |- ( E. v v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 3234	. . . . . 6 |- ( E. v v e. B -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 5498	. . . . . . 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 3243	. . . . 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 5572	. . 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 1398   E.wex 1541   e. wcel 2202   C_ wss 3201   <.cop 3676   X. cxp 4729   ran crn 4732   |` cres 4733   Fn wfn 5328   -->wf 5329   -onto->wfo 5331   ` cfv 5333   1stc1st 6310

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312

This theorem is referenced by:  1stconst  6395