Theorem 1stconst 6118

Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
1stconst	|- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A )

Proof of Theorem 1stconst

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

Step	Hyp	Ref	Expression
1		snmg 3641	. . 3 |- ( B e. V -> E. x x e. { B } )
2		fo1stresm 6059	. . 3 |- ( E. x x e. { B } -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -onto-> A )
3	1, 2	syl 14	. 2 |- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -onto-> A )
4		moeq 2859	. . . . . 6 |- E* x x = <. y , B >.
5	4	moani 2069	. . . . 5 |- E* x ( y e. A /\ x = <. y , B >. )
6		vex 2689	. . . . . . . 8 |- y e. _V
7	6	brres 4825	. . . . . . 7 |- ( x ( 1st |` ( A X. { B } ) ) y <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
8		fo1st 6055	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
9		fofn 5347	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 1st Fn _V
11		vex 2689	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5462	. . . . . . . . . 10 |- ( ( 1st Fn _V /\ x e. _V ) -> ( ( 1st ` x ) = y <-> x 1st y ) )
13	10, 11, 12	mp2an 422	. . . . . . . . 9 |- ( ( 1st ` x ) = y <-> x 1st y )
14	13	anbi1i 453	. . . . . . . 8 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
15		elxp7 6068	. . . . . . . . . . 11 |- ( x e. ( A X. { B } ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) )
16		eleq1 2202	. . . . . . . . . . . . . . 15 |- ( ( 1st ` x ) = y -> ( ( 1st ` x ) e. A <-> y e. A ) )
17	16	biimpa 294	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( 1st ` x ) e. A ) -> y e. A )
18	17	adantrr 470	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) -> y e. A )
19	18	adantrl 469	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> y e. A )
20		elsni 3545	. . . . . . . . . . . . . 14 |- ( ( 2nd ` x ) e. { B } -> ( 2nd ` x ) = B )
21		eqopi 6070	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = y /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
22	21	an12s 554	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
23	20, 22	sylanr2 402	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) e. { B } ) ) -> x = <. y , B >. )
24	23	adantrrl 477	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> x = <. y , B >. )
25	19, 24	jca 304	. . . . . . . . . . 11 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> ( y e. A /\ x = <. y , B >. ) )
26	15, 25	sylan2b 285	. . . . . . . . . 10 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) -> ( y e. A /\ x = <. y , B >. ) )
27	26	adantl 275	. . . . . . . . 9 |- ( ( B e. V /\ ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) ) -> ( y e. A /\ x = <. y , B >. ) )
28		simprr 521	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x = <. y , B >. )
29	28	fveq2d 5425	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = ( 1st ` <. y , B >. ) )
30		simprl 520	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> y e. A )
31		simpl 108	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> B e. V )
32		op1stg 6048	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. V ) -> ( 1st ` <. y , B >. ) = y )
33	30, 31, 32	syl2anc 408	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` <. y , B >. ) = y )
34	29, 33	eqtrd 2172	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = y )
35		snidg 3554	. . . . . . . . . . . . 13 |- ( B e. V -> B e. { B } )
36	35	adantr 274	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> B e. { B } )
37		opelxpi 4571	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. { B } ) -> <. y , B >. e. ( A X. { B } ) )
38	30, 36, 37	syl2anc 408	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> <. y , B >. e. ( A X. { B } ) )
39	28, 38	eqeltrd 2216	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x e. ( A X. { B } ) )
40	34, 39	jca 304	. . . . . . . . 9 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) )
41	27, 40	impbida 585	. . . . . . . 8 |- ( B e. V -> ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) <-> ( y e. A /\ x = <. y , B >. ) ) )
42	14, 41	syl5bbr 193	. . . . . . 7 |- ( B e. V -> ( ( x 1st y /\ x e. ( A X. { B } ) ) <-> ( y e. A /\ x = <. y , B >. ) ) )
43	7, 42	syl5bb 191	. . . . . 6 |- ( B e. V -> ( x ( 1st |` ( A X. { B } ) ) y <-> ( y e. A /\ x = <. y , B >. ) ) )
44	43	mobidv 2035	. . . . 5 |- ( B e. V -> ( E* x x ( 1st |` ( A X. { B } ) ) y <-> E* x ( y e. A /\ x = <. y , B >. ) ) )
45	5, 44	mpbiri 167	. . . 4 |- ( B e. V -> E* x x ( 1st |` ( A X. { B } ) ) y )
46	45	alrimiv 1846	. . 3 |- ( B e. V -> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
47		funcnv2 5183	. . 3 |- ( Fun `' ( 1st |` ( A X. { B } ) ) <-> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
48	46, 47	sylibr 133	. 2 |- ( B e. V -> Fun `' ( 1st |` ( A X. { B } ) ) )
49		dff1o3 5373	. 2 |- ( ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A <-> ( ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -onto-> A /\ Fun `' ( 1st |` ( A X. { B } ) ) ) )
50	3, 48, 49	sylanbrc 413	1 |- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   E*wmo 2000   _Vcvv 2686   {csn 3527   <.cop 3530   class class class wbr 3929   X. cxp 4537   `'ccnv 4538   |` cres 4541   Fun wfun 5117   Fn wfn 5118   -onto->wfo 5121   -1-1-onto->wf1o 5122   ` cfv 5123   1stc1st 6036   2ndc2nd 6037

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039

This theorem is referenced by: (None)