Theorem 2ndconst 6201

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 3701	. . 3 |- ( A e. V -> E. x x e. { A } )
2		fo2ndresm 6141	. . 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 2905	. . . . . 6 |- E* x x = <. A , y >.
5	4	moani 2089	. . . . 5 |- E* x ( y e. B /\ x = <. A , y >. )
6		vex 2733	. . . . . . . 8 |- y e. _V
7	6	brres 4897	. . . . . . 7 |- ( x ( 2nd |` ( { A } X. B ) ) y <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
8		fo2nd 6137	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
9		fofn 5422	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
11		vex 2733	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5537	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ x e. _V ) -> ( ( 2nd ` x ) = y <-> x 2nd y ) )
13	10, 11, 12	mp2an 424	. . . . . . . . 9 |- ( ( 2nd ` x ) = y <-> x 2nd y )
14	13	anbi1i 455	. . . . . . . 8 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
15		elxp7 6149	. . . . . . . . . . 11 |- ( x e. ( { A } X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) )
16		eleq1 2233	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) -> y e. B )
19	18	adantrl 475	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> y e. B )
20		elsni 3601	. . . . . . . . . . . . . 14 |- ( ( 1st ` x ) e. { A } -> ( 1st ` x ) = A )
21		eqopi 6151	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = A /\ ( 2nd ` x ) = y ) ) -> x = <. A , y >. )
22	21	ancom2s 561	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 2nd ` x ) = y /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
23	22	an12s 560	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 5496	. . . . . . . . . . . 12 |- ( x = <. A , y >. -> ( 2nd ` x ) = ( 2nd ` <. A , y >. ) )
30		op2ndg 6130	. . . . . . . . . . . . 13 |- ( ( A e. V /\ y e. _V ) -> ( 2nd ` <. A , y >. ) = y )
31	6, 30	mpan2 423	. . . . . . . . . . . 12 |- ( A e. V -> ( 2nd ` <. A , y >. ) = y )
32	29, 31	sylan9eqr 2225	. . . . . . . . . . 11 |- ( ( A e. V /\ x = <. A , y >. ) -> ( 2nd ` x ) = y )
33	32	adantrl 475	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( 2nd ` x ) = y )
34		simprr 527	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x = <. A , y >. )
35		snidg 3612	. . . . . . . . . . . . 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 526	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> y e. B )
38		opelxpi 4643	. . . . . . . . . . . 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 2247	. . . . . . . . . 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 591	. . . . . . . 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 2055	. . . . 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 1867	. . 3 |- ( A e. V -> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
48		funcnv2 5258	. . 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 5448	. 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 415	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 1346   = wceq 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   _Vcvv 2730   {csn 3583   <.cop 3586   class class class wbr 3989   X. cxp 4609   `'ccnv 4610   |` cres 4613   Fun wfun 5192   Fn wfn 5193   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198   1stc1st 6117   2ndc2nd 6118

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-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by:  xpfi  6907  fsum2dlemstep  11397  fprod2dlemstep  11585