Theorem 1stconst 6395

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 3794	. . 3 |- ( B e. V -> E. x x e. { B } )
2		fo1stresm 6333	. . 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 2982	. . . . . 6 |- E* x x = <. y , B >.
5	4	moani 2150	. . . . 5 |- E* x ( y e. A /\ x = <. y , B >. )
6		vex 2806	. . . . . . . 8 |- y e. _V
7	6	brres 5025	. . . . . . 7 |- ( x ( 1st |` ( A X. { B } ) ) y <-> ( x 1st y /\ x e. ( A X. { B } ) ) )
8		fo1st 6329	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
9		fofn 5570	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- 1st Fn _V
11		vex 2806	. . . . . . . . . 10 |- x e. _V
12		fnbrfvb 5693	. . . . . . . . . 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 6342	. . . . . . . . . . 11 |- ( x e. ( A X. { B } ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. { B } ) ) )
16		eleq1 2294	. . . . . . . . . . . . . . 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 3691	. . . . . . . . . . . . . 14 |- ( ( 2nd ` x ) e. { B } -> ( 2nd ` x ) = B )
21		eqopi 6344	. . . . . . . . . . . . . . 15 |- ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) = y /\ ( 2nd ` x ) = B ) ) -> x = <. y , B >. )
22	21	an12s 567	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . 12 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> x = <. y , B >. )
29	28	fveq2d 5652	. . . . . . . . . . 11 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = ( 1st ` <. y , B >. ) )
30		simprl 531	. . . . . . . . . . . 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 6322	. . . . . . . . . . . 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 2264	. . . . . . . . . 10 |- ( ( B e. V /\ ( y e. A /\ x = <. y , B >. ) ) -> ( 1st ` x ) = y )
35		snidg 3702	. . . . . . . . . . . . 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 4763	. . . . . . . . . . . 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 2308	. . . . . . . . . 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 600	. . . . . . . 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 2115	. . . . 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 1922	. . 3 |- ( B e. V -> A. y E* x x ( 1st |` ( A X. { B } ) ) y )
47		funcnv2 5397	. . 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 5598	. 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 1396   = wceq 1398   E.wex 1541   E*wmo 2080   e. wcel 2202   _Vcvv 2803   {csn 3673   <.cop 3676   class class class wbr 4093   X. cxp 4729   `'ccnv 4730   |` cres 4733   Fun wfun 5327   Fn wfn 5328   -onto->wfo 5331   -1-1-onto->wf1o 5332   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)