Theorem op1steq 6351

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 4840	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 3224	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		eqid 2231	. . . . . 6 |- ( 2nd ` A ) = ( 2nd ` A )
4		eqopi 6344	. . . . . 6 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = ( 2nd ` A ) ) ) -> A = <. B , ( 2nd ` A ) >. )
5	3, 4	mpanr2 438	. . . . 5 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = B ) -> A = <. B , ( 2nd ` A ) >. )
6		2ndexg 6340	. . . . . . 7 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) e. _V )
7		opeq2 3868	. . . . . . . . 9 |- ( x = ( 2nd ` A ) -> <. B , x >. = <. B , ( 2nd ` A ) >. )
8	7	eqeq2d 2243	. . . . . . . 8 |- ( x = ( 2nd ` A ) -> ( A = <. B , x >. <-> A = <. B , ( 2nd ` A ) >. ) )
9	8	spcegv 2895	. . . . . . 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 276	. . . . 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 115	. . 3 |- ( A e. ( _V X. _V ) -> ( ( 1st ` A ) = B -> E. x A = <. B , x >. ) )
14		eqop 6349	. . . . 5 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) ) )
15		simpl 109	. . . . 5 |- ( ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) -> ( 1st ` A ) = B )
16	14, 15	biimtrdi 163	. . . 4 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. -> ( 1st ` A ) = B ) )
17	16	exlimdv 1867	. . 3 |- ( A e. ( _V X. _V ) -> ( E. x A = <. B , x >. -> ( 1st ` A ) = B ) )
18	13, 17	impbid 129	. 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 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   <.cop 3676   X. cxp 4729   ` 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-v 2805  df-sbc 3033  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-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-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by:  releldm2  6357