Theorem op1steq 5836

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 4474	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 2996	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		eqid 2082	. . . . . 6 |- ( 2nd ` A ) = ( 2nd ` A )
4		eqopi 5829	. . . . . 6 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = ( 2nd ` A ) ) ) -> A = <. B , ( 2nd ` A ) >. )
5	3, 4	mpanr2 429	. . . . 5 |- ( ( A e. ( _V X. _V ) /\ ( 1st ` A ) = B ) -> A = <. B , ( 2nd ` A ) >. )
6		2ndexg 5826	. . . . . . 7 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) e. _V )
7		opeq2 3579	. . . . . . . . 9 |- ( x = ( 2nd ` A ) -> <. B , x >. = <. B , ( 2nd ` A ) >. )
8	7	eqeq2d 2093	. . . . . . . 8 |- ( x = ( 2nd ` A ) -> ( A = <. B , x >. <-> A = <. B , ( 2nd ` A ) >. ) )
9	8	spcegv 2687	. . . . . . 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 270	. . . . 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 113	. . 3 |- ( A e. ( _V X. _V ) -> ( ( 1st ` A ) = B -> E. x A = <. B , x >. ) )
14		eqop 5834	. . . . 5 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) ) )
15		simpl 107	. . . . 5 |- ( ( ( 1st ` A ) = B /\ ( 2nd ` A ) = x ) -> ( 1st ` A ) = B )
16	14, 15	syl6bi 161	. . . 4 |- ( A e. ( _V X. _V ) -> ( A = <. B , x >. -> ( 1st ` A ) = B ) )
17	16	exlimdv 1741	. . 3 |- ( A e. ( _V X. _V ) -> ( E. x A = <. B , x >. -> ( 1st ` A ) = B ) )
18	13, 17	impbid 127	. 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 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   _Vcvv 2602   <.cop 3409   X. cxp 4369   ` cfv 4932   1stc1st 5796   2ndc2nd 5797

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-1st 5798  df-2nd 5799

This theorem is referenced by:  releldm2  5842