Theorem 2ndconst 6190

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 3694	. . 3 |- ( A e. V -> E. x x e. { A } )
2		fo2ndresm 6130	. . 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 2901	. . . . . 6 |- E* x x = <. A , y >.
5	4	moani 2084	. . . . 5 |- E* x ( y e. B /\ x = <. A , y >. )
6		vex 2729	. . . . . . . 8 |- y e. _V
7	6	brres 4890	. . . . . . 7 |- ( x ( 2nd |` ( { A } X. B ) ) y <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
8		fo2nd 6126	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
9		fofn 5412	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
11		vex 2729	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5527	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ x e. _V ) -> ( ( 2nd ` x ) = y <-> x 2nd y ) )
13	10, 11, 12	mp2an 423	. . . . . . . . 9 |- ( ( 2nd ` x ) = y <-> x 2nd y )
14	13	anbi1i 454	. . . . . . . 8 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
15		elxp7 6138	. . . . . . . . . . 11 |- ( x e. ( { A } X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) )
16		eleq1 2229	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) -> y e. B )
19	18	adantrl 470	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> y e. B )
20		elsni 3594	. . . . . . . . . . . . . 14 |- ( ( 1st ` x ) e. { A } -> ( 1st ` x ) = A )
21		eqopi 6140	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = A /\ ( 2nd ` x ) = y ) ) -> x = <. A , y >. )
22	21	ancom2s 556	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 2nd ` x ) = y /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
23	22	an12s 555	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
24	20, 23	sylanr2 403	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) e. { A } ) ) -> x = <. A , y >. )
25	24	adantrrr 479	. . . . . . . . . . . 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 5486	. . . . . . . . . . . 12 |- ( x = <. A , y >. -> ( 2nd ` x ) = ( 2nd ` <. A , y >. ) )
30		op2ndg 6119	. . . . . . . . . . . . 13 |- ( ( A e. V /\ y e. _V ) -> ( 2nd ` <. A , y >. ) = y )
31	6, 30	mpan2 422	. . . . . . . . . . . 12 |- ( A e. V -> ( 2nd ` <. A , y >. ) = y )
32	29, 31	sylan9eqr 2221	. . . . . . . . . . 11 |- ( ( A e. V /\ x = <. A , y >. ) -> ( 2nd ` x ) = y )
33	32	adantrl 470	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( 2nd ` x ) = y )
34		simprr 522	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x = <. A , y >. )
35		snidg 3605	. . . . . . . . . . . . 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 521	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> y e. B )
38		opelxpi 4636	. . . . . . . . . . . 12 |- ( ( A e. { A } /\ y e. B ) -> <. A , y >. e. ( { A } X. B ) )
39	36, 37, 38	syl2anc 409	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> <. A , y >. e. ( { A } X. B ) )
40	34, 39	eqeltrd 2243	. . . . . . . . . 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 586	. . . . . . . 8 |- ( A e. V -> ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( y e. B /\ x = <. A , y >. ) ) )
43	14, 42	bitr3id 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 2050	. . . . 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 1862	. . 3 |- ( A e. V -> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
48		funcnv2 5248	. . 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 5438	. 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 414	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 1341   = wceq 1343   E.wex 1480   E*wmo 2015   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   class class class wbr 3982   X. cxp 4602   `'ccnv 4603   |` cres 4606   Fun wfun 5182   Fn wfn 5183   -onto->wfo 5186   -1-1-onto->wf1o 5187   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  xpfi  6895  fsum2dlemstep  11375  fprod2dlemstep  11563