Theorem xp2nd 5813

Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
xp2nd	|- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

Proof of Theorem xp2nd

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4380	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2604	. . . . . . 7 |- b e. _V
3		vex 2604	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 5796	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2147	. . . . 5 |- ( A = <. b , c >. -> ( ( 2nd ` A ) e. C <-> c e. C ) )
6	5	biimpar 291	. . . 4 |- ( ( A = <. b , c >. /\ c e. C ) -> ( 2nd ` A ) e. C )
7	6	adantrl 461	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
8	7	exlimivv 1817	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
9	1, 8	sylbi 119	1 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   <.cop 3401   X. cxp 4361   ` cfv 4922   2ndc2nd 5786

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-2nd 5788

This theorem is referenced by:  dfplpq2  6544  dfmpq2  6545  enqbreq2  6547  enqdc1  6552  mulpipq2  6561  preqlu  6662  elnp1st2nd  6666  cauappcvgprlemladd  6848  elreal2  6999  cnref1o  8733  frecuzrdgrrn  9410  frec2uzrdg  9411  frecuzrdgfn  9414  frecuzrdgcl  9415  frecuzrdgsuc  9417  eucalgval  10436  eucalginv  10438  eucalglt  10439  eucialgcvga  10440  eucialg  10441  sqpweven  10553  2sqpwodd  10554