Theorem elxp6 4086

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 3439.

Assertion

Ref	Expression
elxp6	|- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))

Proof of Theorem elxp6

Step	Hyp	Ref	Expression
1		elxp4 3439	. 2 |- (A e. (B X. C) <-> (A = <.U.dom { A}, U.ran { A}>. /\ (U.dom { A} e. B /\ U.ran { A} e. C)))
2		1stval 4065	. . . . 5 |- (1st` A) = U.dom { A}
3		2ndval 4066	. . . . 5 |- (2nd` A) = U.ran { A}
4	2, 3	opeq12i 2483	. . . 4 |- <.(1st` A), (2nd` A)>. = <.U.dom { A}, U.ran { A}>.
5	4	eqeq2i 1477	. . 3 |- (A = <.(1st` A), (2nd` A)>. <-> A = <.U.dom { A}, U.ran { A}>.)
6	2	eleq1i 1529	. . . 4 |- ((1st` A) e. B <-> U.dom { A} e. B)
7	3	eleq1i 1529	. . . 4 |- ((2nd` A) e. C <-> U.ran { A} e. C)
8	6, 7	anbi12i 481	. . 3 |- (((1st` A) e. B /\ (2nd` A) e. C) <-> (U.dom { A} e. B /\ U.ran { A} e. C))
9	5, 8	anbi12i 481	. 2 |- ((A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)) <-> (A = <.U.dom { A}, U.ran { A}>. /\ (U.dom { A} e. B /\ U.ran { A} e. C)))
10	1, 9	bitr4 176	1 |- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {csn 2399  <.cop 2401  U.cuni 2493   X. cxp 3158  dom cdm 3160  ran crn 3161  ` cfv 3172  1stc1st 4061  2ndc2nd 4062

This theorem is referenced by:  elxp7 4087  eqop 4088  xpopth 4090  1st2nd 4092  ruclem13 7465  ruclem23 7475  xplmi 7907  xplmi2 7908  bopcnlem2 7916  bopcnlem3 7917  bcthlem4 7936  bcthlem14 7946  sspval 8316

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-1st 4063  df-2nd 4064

Copyright terms: Public domain