Theorem 2ndconst 7266

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		snnzg 4308	. . 3 |- ( A e. V -> { A } =/= (/) )
2		fo2ndres 7193	. . 3 |- ( { A } =/= (/) -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -onto-> B )
3	1, 2	syl 17	. 2 |- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -onto-> B )
4		moeq 3382	. . . . . 6 |- E* x x = <. A , y >.
5	4	moani 2525	. . . . 5 |- E* x ( y e. B /\ x = <. A , y >. )
6		vex 3203	. . . . . . . 8 |- y e. _V
7	6	brres 5402	. . . . . . 7 |- ( x ( 2nd |` ( { A } X. B ) ) y <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
8		fo2nd 7189	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
9		fofn 6117	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
11		vex 3203	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 6236	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ x e. _V ) -> ( ( 2nd ` x ) = y <-> x 2nd y ) )
13	10, 11, 12	mp2an 708	. . . . . . . . 9 |- ( ( 2nd ` x ) = y <-> x 2nd y )
14	13	anbi1i 731	. . . . . . . 8 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( x 2nd y /\ x e. ( { A } X. B ) ) )
15		elxp7 7201	. . . . . . . . . . 11 |- ( x e. ( { A } X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) )
16		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` x ) = y -> ( ( 2nd ` x ) e. B <-> y e. B ) )
17	16	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` x ) = y /\ ( 2nd ` x ) e. B ) -> y e. B )
18	17	adantrl 752	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) -> y e. B )
19	18	adantrl 752	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> y e. B )
20		elsni 4194	. . . . . . . . . . . . . 14 |- ( ( 1st ` x ) e. { A } -> ( 1st ` x ) = A )
21		eqopi 7202	. . . . . . . . . . . . . . . 16 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = A /\ ( 2nd ` x ) = y ) ) -> x = <. A , y >. )
22	21	ancom2s 844	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 2nd ` x ) = y /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
23	22	an12s 843	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) = A ) ) -> x = <. A , y >. )
24	20, 23	sylanr2 685	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 1st ` x ) e. { A } ) ) -> x = <. A , y >. )
25	24	adantrrr 761	. . . . . . . . . . . 12 |- ( ( ( 2nd ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. { A } /\ ( 2nd ` x ) e. B ) ) ) -> x = <. A , y >. )
26	19, 25	jca 554	. . . . . . . . . . 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 492	. . . . . . . . . 10 |- ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) -> ( y e. B /\ x = <. A , y >. ) )
28	27	adantl 482	. . . . . . . . 9 |- ( ( A e. V /\ ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) ) -> ( y e. B /\ x = <. A , y >. ) )
29		fveq2 6191	. . . . . . . . . . . 12 |- ( x = <. A , y >. -> ( 2nd ` x ) = ( 2nd ` <. A , y >. ) )
30		op2ndg 7181	. . . . . . . . . . . . 13 |- ( ( A e. V /\ y e. _V ) -> ( 2nd ` <. A , y >. ) = y )
31	6, 30	mpan2 707	. . . . . . . . . . . 12 |- ( A e. V -> ( 2nd ` <. A , y >. ) = y )
32	29, 31	sylan9eqr 2678	. . . . . . . . . . 11 |- ( ( A e. V /\ x = <. A , y >. ) -> ( 2nd ` x ) = y )
33	32	adantrl 752	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( 2nd ` x ) = y )
34		simprr 796	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x = <. A , y >. )
35		snidg 4206	. . . . . . . . . . . . 13 |- ( A e. V -> A e. { A } )
36	35	adantr 481	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> A e. { A } )
37		simprl 794	. . . . . . . . . . . 12 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> y e. B )
38		opelxpi 5148	. . . . . . . . . . . 12 |- ( ( A e. { A } /\ y e. B ) -> <. A , y >. e. ( { A } X. B ) )
39	36, 37, 38	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> <. A , y >. e. ( { A } X. B ) )
40	34, 39	eqeltrd 2701	. . . . . . . . . 10 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> x e. ( { A } X. B ) )
41	33, 40	jca 554	. . . . . . . . 9 |- ( ( A e. V /\ ( y e. B /\ x = <. A , y >. ) ) -> ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) )
42	28, 41	impbida 877	. . . . . . . 8 |- ( A e. V -> ( ( ( 2nd ` x ) = y /\ x e. ( { A } X. B ) ) <-> ( y e. B /\ x = <. A , y >. ) ) )
43	14, 42	syl5bbr 274	. . . . . . 7 |- ( A e. V -> ( ( x 2nd y /\ x e. ( { A } X. B ) ) <-> ( y e. B /\ x = <. A , y >. ) ) )
44	7, 43	syl5bb 272	. . . . . 6 |- ( A e. V -> ( x ( 2nd |` ( { A } X. B ) ) y <-> ( y e. B /\ x = <. A , y >. ) ) )
45	44	mobidv 2491	. . . . 5 |- ( A e. V -> ( E* x x ( 2nd |` ( { A } X. B ) ) y <-> E* x ( y e. B /\ x = <. A , y >. ) ) )
46	5, 45	mpbiri 248	. . . 4 |- ( A e. V -> E* x x ( 2nd |` ( { A } X. B ) ) y )
47	46	alrimiv 1855	. . 3 |- ( A e. V -> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
48		funcnv2 5957	. . 3 |- ( Fun `' ( 2nd |` ( { A } X. B ) ) <-> A. y E* x x ( 2nd |` ( { A } X. B ) ) y )
49	47, 48	sylibr 224	. 2 |- ( A e. V -> Fun `' ( 2nd |` ( { A } X. B ) ) )
50		dff1o3 6143	. 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 698	1 |- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  curry1  7269  xpfi  8231  fsum2dlem  14501  fprod2dlem  14710  gsum2dlem2  18370  ovoliunlem1  23270  gsummpt2d  29781  fv2ndcnv  31681