Theorem op1steq 6288

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 4801	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 3197	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		eqid 2207	. . . . . 6 |- ( 2nd ` A ) = ( 2nd ` A )
4		eqopi 6281	. . . . . 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 6277	. . . . . . 7 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) e. _V )
7		opeq2 3834	. . . . . . . . 9 |- ( x = ( 2nd ` A ) -> <. B , x >. = <. B , ( 2nd ` A ) >. )
8	7	eqeq2d 2219	. . . . . . . 8 |- ( x = ( 2nd ` A ) -> ( A = <. B , x >. <-> A = <. B , ( 2nd ` A ) >. ) )
9	8	spcegv 2868	. . . . . . 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 6286	. . . . 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 1843	. . 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 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   <.cop 3646   X. cxp 4691   ` cfv 5290   1stc1st 6247   2ndc2nd 6248

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-1st 6249  df-2nd 6250

This theorem is referenced by:  releldm2  6294