Theorem fo1stresm 6059

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 2202	. . 3 |- ( v = y -> ( v e. B <-> y e. B ) )
2	1	cbvexv 1890	. 2 |- ( E. v v e. B <-> E. y y e. B )
3		opelxp 4569	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5445	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) = ( 1st ` <. u , v >. ) )
5		vex 2689	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2689	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op1st 6044	. . . . . . . . . . . 12 |- ( 1st ` <. u , v >. ) = u
8	4, 7	syl6req 2189	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> u = ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) )
9		f1stres 6057	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
10		ffn 5272	. . . . . . . . . . . . 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 5552	. . . . . . . . . . . 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 420	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 1st |` ( A X. B ) ) )
14	8, 13	eqeltrd 2216	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
15	3, 14	sylbir 134	. . . . . . . . 9 |- ( ( u e. A /\ v e. B ) -> u e. ran ( 1st |` ( A X. B ) ) )
16	15	expcom 115	. . . . . . . 8 |- ( v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
17	16	exlimiv 1577	. . . . . . 7 |- ( E. v v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 3103	. . . . . 6 |- ( E. v v e. B -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 5281	. . . . . . 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 310	. . . . 5 |- ( E. v v e. B -> ( ran ( 1st |` ( A X. B ) ) C_ A /\ A C_ ran ( 1st |` ( A X. B ) ) ) )
22		eqss 3112	. . . . 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 133	. . . 4 |- ( E. v v e. B -> ran ( 1st |` ( A X. B ) ) = A )
24	23, 9	jctil 310	. . 3 |- ( E. v v e. B -> ( ( 1st |` ( A X. B ) ) : ( A X. B ) --> A /\ ran ( 1st |` ( A X. B ) ) = A ) )
25		dffo2 5349	. . 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 133	. 2 |- ( E. v v e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )
27	2, 26	sylbir 134	1 |- ( E. y y e. B -> ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   C_ wss 3071   <.cop 3530   X. cxp 4537   ran crn 4540   |` cres 4541   Fn wfn 5118   -->wf 5119   -onto->wfo 5121   ` cfv 5123   1stc1st 6036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038

This theorem is referenced by:  1stconst  6118