Theorem xp2nd 6400

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 4791	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2824	. . . . . . 7 |- b e. _V
3		vex 2824	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 6383	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2307	. . . . 5 |- ( A = <. b , c >. -> ( ( 2nd ` A ) e. C <-> c e. C ) )
6	5	biimpar 297	. . . 4 |- ( ( A = <. b , c >. /\ c e. C ) -> ( 2nd ` A ) e. C )
7	6	adantrl 482	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
8	7	exlimivv 1952	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
9	1, 8	sylbi 121	1 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   <.cop 3712   X. cxp 4772   ` cfv 5377   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-2nd 6375

This theorem is used by:  xpf1o  7144  xpmapenlem  7149  mapunen  7151  opabfi  7247  djuf1olem  7393  exmidapne  7626  cc2lem  7632  dfplpq2  7721  dfmpq2  7722  enqbreq2  7724  enqdc1  7729  mulpipq2  7738  preqlu  7839  elnp1st2nd  7843  cauappcvgprlemladd  8025  elreal2  8197  cnref1o  10051  frecuzrdgrrn  10845  frec2uzrdg  10846  frecuzrdgrcl  10847  frecuzrdgtcl  10849  frecuzrdgsuc  10851  frecuzrdgrclt  10852  frecuzrdgg  10853  frecuzrdgdomlem  10854  frecuzrdgfunlem  10856  frecuzrdgsuctlem  10860  seq3val  10897  seqvalcd  10898  fisumcom2  12205  fprodcom2fi  12393  eucalgval  12832  eucalginv  12834  eucalglt  12835  eucalgcvga  12836  eucalg  12837  sqpweven  12953  2sqpwodd  12954  ctiunctlemudc  13328  xpsff1o  13670  tx1cn  15370  txdis  15378  txhmeo  15420  xmetxp  15608  xmetxpbl  15609  xmettxlem  15610  xmettx  15611