Theorem 2ndconst 6112

Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Assertion

Ref	Expression
2ndconst	|- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B )

Proof of Theorem 2ndconst

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snmg 3636	. . 3 |- ( A e. V -> E. x x e. { A } )
2		fo2ndresm 6053	. . 3 |- ( E. x x e. { A } -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -onto-> B )
3	1, 2	syl 14	. 2 |- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -onto-> B )
4		moeq 2854	. . . . . 6 |- E* x x = <. A , y >.
5	4	moani 2067	. . . . 5 |- E* x ( y e. B /\ x = <. A , y >. )
6		vex 2684	. . . . . . . 8 |- y e. _V
7	6	brres 4820	. . . . . . 7 |- ( x ( 2nd |` ( { A } X. B ) ) y <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
8		fo2nd 6049	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
9		fofn 5342	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
11		vex 2684	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5455	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ x e. _V ) -> ( ( 2nd ` x ) = y <-> x 2nd y ) )
13	10, 11, 12	mp2an 422	. . . . . . . . 9 |- ( ( 2nd ` x ) = y <-> x 2nd y )
14	13	anbi1i 453	. . . . . . . 8 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
15		elxp7 6061	. . . . . . . . . . 11 |- ( x e. ( { A } X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) )
16		eleq1 2200	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` x ) = y -> ( ( 2nd ` x ) e. B <-> y e. B ) )
17	16	biimpa 294	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` x ) = y /\ ( 2nd ` x ) e. B ) -> y e. B )
18	17	adantrl 469	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) -> y e. B )
19	18	adantrl 469	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> y e. B )
20		elsni 3540	. . . . . . . . . . . . . 14 |- ( ( 1st ` x ) e. { A } -> ( 1st ` x ) = A )
21		eqopi 6063	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = A /\ ( 2nd ` x ) = y ) ) -> x = <. A , y >. )
22	21	ancom2s 555	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 2nd ` x ) = y /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
23	22	an12s 554	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
24	20, 23	sylanr2 402	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) e. { A } ) ) -> x = <. A , y >. )
25	24	adantrrr 478	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> x = <. A , y >. )
26	19, 25	jca 304	. . . . . . . . . . 11 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> ( y e. B /\ x = <. A , y >. ) )
27	15, 26	sylan2b 285	. . . . . . . . . 10 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) -> ( y e. B /\ x = <. A , y >. ) )
28	27	adantl 275	. . . . . . . . 9 |- ( ( A e. V /\ ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) ) -> ( y e. B /\ x = <. A , y >. ) )
29		fveq2 5414	. . . . . . . . . . . 12 |- ( x = <. A , y >. -> ( 2nd ` x ) = ( 2nd ` <. A , y >. ) )
30		op2ndg 6042	. . . . . . . . . . . . 13 |- ( ( A e. V /\ y e. _V ) -> ( 2nd ` <. A , y >. ) = y )
31	6, 30	mpan2 421	. . . . . . . . . . . 12 |- ( A e. V -> ( 2nd ` <. A , y >. ) = y )
32	29, 31	sylan9eqr 2192	. . . . . . . . . . 11 |- ( ( A e. V /\ x = <. A , y >. ) -> ( 2nd ` x ) = y )
33	32	adantrl 469	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( 2nd ` x ) = y )
34		simprr 521	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x = <. A , y >. )
35		snidg 3549	. . . . . . . . . . . . 13 |- ( A e. V -> A e. { A } )
36	35	adantr 274	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> A e. { A } )
37		simprl 520	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> y e. B )
38		opelxpi 4566	. . . . . . . . . . . 12 |- ( ( A e. { A } /\ y e. B ) -> <. A , y >. e. ( { A } X. B ) )
39	36, 37, 38	syl2anc 408	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> <. A , y >. e. ( { A } X. B ) )
40	34, 39	eqeltrd 2214	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x e. ( { A } X. B ) )
41	33, 40	jca 304	. . . . . . . . 9 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) )
42	28, 41	impbida 585	. . . . . . . 8 |- ( A e. V -> ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( y e. B /\ x = <. A , y >. ) ) )
43	14, 42	syl5bbr 193	. . . . . . 7 |- ( A e. V -> ( ( x 2nd y /\ x e. ( { A } X. B ) ) <-> ( y e. B /\ x = <. A , y >. ) ) )
44	7, 43	syl5bb 191	. . . . . 6 |- ( A e. V -> ( x ( 2nd |` ( { A } X. B ) ) y <-> ( y e. B /\ x = <. A , y >. ) ) )
45	44	mobidv 2033	. . . . 5 |- ( A e. V -> ( E* x x ( 2nd |` ( { A } X. B ) ) y <-> E* x ( y e. B /\ x = <. A , y >. ) ) )
46	5, 45	mpbiri 167	. . . 4 |- ( A e. V -> E* x x ( 2nd |` ( { A } X. B ) ) y )
47	46	alrimiv 1846	. . 3 |- ( A e. V -> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
48		funcnv2 5178	. . 3 |- ( Fun `' ( 2nd |` ( { A } X. B ) ) <-> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
49	47, 48	sylibr 133	. 2 |- ( A e. V -> Fun `' ( 2nd |` ( { A } X. B ) ) )
50		dff1o3 5366	. 2 |- ( ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B <-> ( ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -onto-> B /\ Fun `' ( 2nd |` ( { A } X. B ) ) ) )
51	3, 49, 50	sylanbrc 413	1 |- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   E*wmo 1998   _Vcvv 2681   {csn 3522   <.cop 3525   class class class wbr 3924   X. cxp 4532   `'ccnv 4533   |` cres 4536   Fun wfun 5112   Fn wfn 5113   -onto->wfo 5116   -1-1-onto->wf1o 5117   ` cfv 5118   1stc1st 6029   2ndc2nd 6030

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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by:  xpfi  6811  fsum2dlemstep  11196