Theorem op1steq 6085

Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
op1steq	|- ( A e. ( V X. W ) -> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem op1steq

Step	Hyp	Ref	Expression
1		xpss 4655	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 3098	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		eqid 2140	. . . . . 6 |- ( 2nd ` A ) = ( 2nd ` A )
4		eqopi 6078	. . . . . 6 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = ( 2nd ` A ) ) ) -> A = <. B , ( 2nd ` A ) >. )
5	3, 4	mpanr2 435	. . . . 5 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = B ) -> A = <. B , ( 2nd ` A ) >. )
6		2ndexg 6074	. . . . . . 7 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) e. _V )
7		opeq2 3714	. . . . . . . . 9 |- ( x = ( 2nd ` A ) -> <. B , x >. = <. B , ( 2nd ` A ) >. )
8	7	eqeq2d 2152	. . . . . . . 8 |- ( x = ( 2nd ` A ) -> ( A = <. B , x >. <-> A = <. B , ( 2nd ` A ) >. ) )
9	8	spcegv 2777	. . . . . . 7 |- ( ( 2nd ` A ) e. _V -> ( A = <. B , ( 2nd ` A ) >. -> E. x A = <. B , x >. ) )
10	6, 9	syl 14	. . . . . 6 |- ( A e. ( _V X. _V ) -> ( A = <. B , ( 2nd ` A ) >. -> E. x A = <. B , x >. ) )
11	10	adantr 274	. . . . 5 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = B ) -> ( A = <. B , ( 2nd ` A ) >. -> E. x A = <. B , x >. ) )
12	5, 11	mpd 13	. . . 4 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = B ) -> E. x A = <. B , x >. )
13	12	ex 114	. . 3 |- ( A e. ( _V X. _V ) -> ( ( 1st ` A ) = B -> E. x A = <. B , x >. ) )
14		eqop 6083	. . . . 5 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) ) )
15		simpl 108	. . . . 5 |- ( ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) -> ( 1st ` A ) = B )
16	14, 15	syl6bi 162	. . . 4 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. -> ( 1st ` A ) = B ) )
17	16	exlimdv 1792	. . 3 |- ( A e. ( _V X. _V ) -> ( E. x A = <. B , x >. -> ( 1st ` A ) = B ) )
18	13, 17	impbid 128	. 2 |- ( A e. ( _V X. _V ) -> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) )
19	2, 18	syl 14	1 |- ( A e. ( V X. W ) -> ( ( 1st ` A ) = B <-> E. x A = <. B , x >. ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   <.cop 3535   X. cxp 4545   ` cfv 5131   1stc1st 6044   2ndc2nd 6045

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by:  releldm2  6091