Theorem fo1stresm 6140

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 2233	. . 3 |- ( v = y -> ( v e. B <-> y e. B ) )
2	1	cbvexv 1911	. 2 |- ( E. v v e. B <-> E. y y e. B )
3		opelxp 4641	. . . . . . . . . 10 |- ( <. u , v >. e. ( A X. B ) <-> ( u e. A /\ v e. B ) )
4		fvres 5520	. . . . . . . . . . . 12 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) = ( 1st ` <. u , v >. ) )
5		vex 2733	. . . . . . . . . . . . 13 |- u e. _V
6		vex 2733	. . . . . . . . . . . . 13 |- v e. _V
7	5, 6	op1st 6125	. . . . . . . . . . . 12 |- ( 1st ` <. u , v >. ) = u
8	4, 7	eqtr2di 2220	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> u = ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) )
9		f1stres 6138	. . . . . . . . . . . . 13 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A
10		ffn 5347	. . . . . . . . . . . . 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 5628	. . . . . . . . . . . 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 422	. . . . . . . . . . 11 |- ( <. u , v >. e. ( A X. B ) -> ( ( 1st |` ( A X. B ) ) ` <. u , v >. ) e. ran ( 1st |` ( A X. B ) ) )
14	8, 13	eqeltrd 2247	. . . . . . . . . 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 1591	. . . . . . 7 |- ( E. v v e. B -> ( u e. A -> u e. ran ( 1st |` ( A X. B ) ) ) )
18	17	ssrdv 3153	. . . . . 6 |- ( E. v v e. B -> A C_ ran ( 1st |` ( A X. B ) ) )
19		frn 5356	. . . . . . 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 3162	. . . . 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 5424	. . 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 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   <.cop 3586   X. cxp 4609   ran crn 4612   |` cres 4613   Fn wfn 5193   -->wf 5194   -onto->wfo 5196   ` cfv 5198   1stc1st 6117

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119

This theorem is referenced by:  1stconst  6200