Theorem 1stconst 6224

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 3712	. . 3 |- ( B e. V -> E. x x e. { B } )
2		fo1stresm 6164	. . 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 2914	. . . . . 6 |- E* x x = <. y , B >.
5	4	moani 2096	. . . . 5 |- E* x ( y e. A /\ x = <. y , B >. )
6		vex 2742	. . . . . . . 8 |- y e. _V
7	6	brres 4915	. . . . . . 7 |- ( x ( 1st |` ( A X. { B } ) ) y <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
8		fo1st 6160	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
9		fofn 5442	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 1st Fn _V
11		vex 2742	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5558	. . . . . . . . . 10 |- ( ( 1st Fn _V /\ x e. _V ) -> ( ( 1st ` x ) = y <-> x 1st y ) )
13	10, 11, 12	mp2an 426	. . . . . . . . 9 |- ( ( 1st ` x ) = y <-> x 1st y )
14	13	anbi1i 458	. . . . . . . 8 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
15		elxp7 6173	. . . . . . . . . . 11 |- ( x e. ( A X. { B } ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) )
16		eleq1 2240	. . . . . . . . . . . . . . 15 |- ( ( 1st ` x ) = y -> ( ( 1st ` x ) e. A <-> y e. A ) )
17	16	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( 1st ` x ) e. A ) -> y e. A )
18	17	adantrr 479	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) -> y e. A )
19	18	adantrl 478	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> y e. A )
20		elsni 3612	. . . . . . . . . . . . . 14 |- ( ( 2nd ` x ) e. { B } -> ( 2nd ` x ) = B )
21		eqopi 6175	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = y /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
22	21	an12s 565	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
23	20, 22	sylanr2 405	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( 2nd ` x ) e. { B } ) ) -> x = <. y , B >. )
24	23	adantrrl 486	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = y /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) ) -> x = <. y , B >. )
25	19, 24	jca 306	. . . . . . . . . . 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 287	. . . . . . . . . 10 |- ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) -> ( y e. A /\ x = <. y , B >. ) )
27	26	adantl 277	. . . . . . . . 9 |- ( ( B e. V /\ ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) ) -> ( y e. A /\ x = <. y , B >. ) )
28		simprr 531	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x = <. y , B >. )
29	28	fveq2d 5521	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = ( 1st ` <. y , B >. ) )
30		simprl 529	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> y e. A )
31		simpl 109	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> B e. V )
32		op1stg 6153	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. V ) -> ( 1st ` <. y , B >. ) = y )
33	30, 31, 32	syl2anc 411	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` <. y , B >. ) = y )
34	29, 33	eqtrd 2210	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = y )
35		snidg 3623	. . . . . . . . . . . . 13 |- ( B e. V -> B e. { B } )
36	35	adantr 276	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> B e. { B } )
37		opelxpi 4660	. . . . . . . . . . . 12 |- ( ( y e. A /\ B e. { B } ) -> <. y , B >. e. ( A X. { B } ) )
38	30, 36, 37	syl2anc 411	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> <. y , B >. e. ( A X. { B } ) )
39	28, 38	eqeltrd 2254	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x e. ( A X. { B } ) )
40	34, 39	jca 306	. . . . . . . . 9 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) )
41	27, 40	impbida 596	. . . . . . . 8 |- ( B e. V -> ( ( ( 1st ` x ) = y /\ x e. ( A X. { B } ) ) <-> ( y e. A /\ x = <. y , B >. ) ) )
42	14, 41	bitr3id 194	. . . . . . 7 |- ( B e. V -> ( ( x 1st y /\ x e. ( A X. { B } ) ) <-> ( y e. A /\ x = <. y , B >. ) ) )
43	7, 42	bitrid 192	. . . . . 6 |- ( B e. V -> ( x ( 1st |` ( A X. { B } ) ) y <-> ( y e. A /\ x = <. y , B >. ) ) )
44	43	mobidv 2062	. . . . 5 |- ( B e. V -> ( E* x x ( 1st |` ( A X. { B } ) ) y <-> E* x ( y e. A /\ x = <. y , B >. ) ) )
45	5, 44	mpbiri 168	. . . 4 |- ( B e. V -> E* x x ( 1st |` ( A X. { B } ) ) y )
46	45	alrimiv 1874	. . 3 |- ( B e. V -> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
47		funcnv2 5278	. . 3 |- ( Fun `' ( 1st |` ( A X. { B } ) ) <-> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
48	46, 47	sylibr 134	. 2 |- ( B e. V -> Fun `' ( 1st |` ( A X. { B } ) ) )
49		dff1o3 5469	. 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 417	1 |- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   E*wmo 2027   e. wcel 2148   _Vcvv 2739   {csn 3594   <.cop 3597   class class class wbr 4005   X. cxp 4626   `'ccnv 4627   |` cres 4630   Fun wfun 5212   Fn wfn 5213   -onto->wfo 5216   -1-1-onto->wf1o 5217   ` cfv 5218   1stc1st 6141   2ndc2nd 6142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144

This theorem is referenced by: (None)