Theorem xp2nd 6265

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 4700	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2776	. . . . . . 7 |- b e. _V
3		vex 2776	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 6248	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2275	. . . . 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 478	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
8	7	exlimivv 1921	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2177   <.cop 3641   X. cxp 4681   ` cfv 5280   2ndc2nd 6238

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-2nd 6240

This theorem is referenced by:  xpf1o  6956  xpmapenlem  6961  opabfi  7050  djuf1olem  7170  exmidapne  7392  cc2lem  7398  dfplpq2  7487  dfmpq2  7488  enqbreq2  7490  enqdc1  7495  mulpipq2  7504  preqlu  7605  elnp1st2nd  7609  cauappcvgprlemladd  7791  elreal2  7963  cnref1o  9792  frecuzrdgrrn  10575  frec2uzrdg  10576  frecuzrdgrcl  10577  frecuzrdgtcl  10579  frecuzrdgsuc  10581  frecuzrdgrclt  10582  frecuzrdgg  10583  frecuzrdgdomlem  10584  frecuzrdgfunlem  10586  frecuzrdgsuctlem  10590  seq3val  10627  seqvalcd  10628  fisumcom2  11824  fprodcom2fi  12012  eucalgval  12451  eucalginv  12453  eucalglt  12454  eucalgcvga  12455  eucalg  12456  sqpweven  12572  2sqpwodd  12573  ctiunctlemudc  12883  xpsff1o  13256  tx1cn  14816  txdis  14824  txhmeo  14866  xmetxp  15054  xmetxpbl  15055  xmettxlem  15056  xmettx  15057