Theorem xp2nd 5793

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 4362	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2560	. . . . . . 7 |- b e. _V
3		vex 2560	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 5776	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2106	. . . . 5 |- ( A = <. b , c >. -> ( ( 2nd ` A ) e. C <-> c e. C ) )
6	5	biimpar 281	. . . 4 |- ( ( A = <. b , c >. /\ c e. C ) -> ( 2nd ` A ) e. C )
7	6	adantrl 447	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
8	7	exlimivv 1776	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
9	1, 8	sylbi 114	1 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   X. cxp 4343   ` cfv 4902   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768

This theorem is referenced by:  dfplpq2  6452  dfmpq2  6453  enqbreq2  6455  enqdc1  6460  mulpipq2  6469  preqlu  6570  elnp1st2nd  6574  cauappcvgprlemladd  6756  elreal2  6907  cnref1o  8582  frecuzrdgrrn  9194  frec2uzrdg  9195  frecuzrdgfn  9198  frecuzrdgcl  9199  frecuzrdgsuc  9201