Theorem op1steq 6237

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 4771	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 3179	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		eqid 2196	. . . . . 6 |- ( 2nd ` A ) = ( 2nd ` A )
4		eqopi 6230	. . . . . 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 6226	. . . . . . 7 |- ( A e. ( _V X. _V ) -> ( 2nd ` A ) e. _V )
7		opeq2 3809	. . . . . . . . 9 |- ( x = ( 2nd ` A ) -> <. B , x >. = <. B , ( 2nd ` A ) >. )
8	7	eqeq2d 2208	. . . . . . . 8 |- ( x = ( 2nd ` A ) -> ( A = <. B , x >. <-> A = <. B , ( 2nd ` A ) >. ) )
9	8	spcegv 2852	. . . . . . 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 6235	. . . . 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 1833	. . 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 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   <.cop 3625   X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by:  releldm2  6243