Theorem 1stconst 6000

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 3564	. . 3 |- ( B e. V -> E. x x e. { B } )
2		fo1stresm 5946	. . 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 2791	. . . . . 6 |- E* x x = <. y , B >.
5	4	moani 2019	. . . . 5 |- E* x ( y e. A /\ x = <. y , B >. )
6		vex 2623	. . . . . . . 8 |- y e. _V
7	6	brres 4732	. . . . . . 7 |- ( x ( 1st |` ( A X. { B } ) ) y <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
8		fo1st 5942	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
9		fofn 5248	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
10	8, 9	ax-mp 7	. . . . . . . . . 10 |- 1st Fn _V
11		vex 2623	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5358	. . . . . . . . . 10 |- ( ( 1st Fn _V /\ x e. _V ) -> ( ( 1st ` x ) = y <-> x 1st y ) )
13	10, 11, 12	mp2an 418	. . . . . . . . 9 |- ( ( 1st ` x ) = y <-> x 1st y )
14	13	anbi1i 447	. . . . . . . 8 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
15		elxp7 5955	. . . . . . . . . . 11 |- ( x e. ( A X. { B } ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) )
16		eleq1 2151	. . . . . . . . . . . . . . 15 |- ( ( 1st ` x ) = y -> ( ( 1st ` x ) e. A <-> y e. A ) )
17	16	biimpa 291	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( 1st ` x ) e. A ) -> y e. A )
18	17	adantrr 464	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) -> y e. A )
19	18	adantrl 463	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> y e. A )
20		elsni 3468	. . . . . . . . . . . . . 14 |- ( ( 2nd ` x ) e. { B } -> ( 2nd ` x ) = B )
21		eqopi 5956	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = y /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
22	21	an12s 533	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
23	20, 22	sylanr2 398	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) e. { B } ) ) -> x = <. y , B >. )
24	23	adantrrl 471	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> x = <. y , B >. )
25	19, 24	jca 301	. . . . . . . . . . 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 282	. . . . . . . . . 10 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) -> ( y e. A /\ x = <. y , B >. ) )
27	26	adantl 272	. . . . . . . . 9 |- ( ( B e. V /\ ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) ) -> ( y e. A /\ x = <. y , B >. ) )
28		simprr 500	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x = <. y , B >. )
29	28	fveq2d 5322	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = ( 1st ` <. y , B >. ) )
30		simprl 499	. . . . . . . . . . . 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 5935	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. V ) -> ( 1st ` <. y , B >. ) = y )
33	30, 31, 32	syl2anc 404	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` <. y , B >. ) = y )
34	29, 33	eqtrd 2121	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = y )
35		snidg 3477	. . . . . . . . . . . . 13 |- ( B e. V -> B e. { B } )
36	35	adantr 271	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> B e. { B } )
37		opelxpi 4483	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. { B } ) -> <. y , B >. e. ( A X. { B } ) )
38	30, 36, 37	syl2anc 404	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> <. y , B >. e. ( A X. { B } ) )
39	28, 38	eqeltrd 2165	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x e. ( A X. { B } ) )
40	34, 39	jca 301	. . . . . . . . 9 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) )
41	27, 40	impbida 564	. . . . . . . 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 1985	. . . . 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 1803	. . 3 |- ( B e. V -> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
47		funcnv2 5087	. . 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 5272	. 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 409	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 1288   = wceq 1290   E.wex 1427   e. wcel 1439   E*wmo 1950   _Vcvv 2620   {csn 3450   <.cop 3453   class class class wbr 3851   X. cxp 4450   `'ccnv 4451   |` cres 4454   Fun wfun 5022   Fn wfn 5023   -onto->wfo 5026   -1-1-onto->wf1o 5027   ` cfv 5028   1stc1st 5923   2ndc2nd 5924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1st 5925  df-2nd 5926

This theorem is referenced by: (None)