Theorem elxp6 5041

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4379.

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 4379	. 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 5022	. . . . 5 |- (1st` A) = U.dom { A}
3		2ndval 5023	. . . . 5 |- (2nd` A) = U.ran { A}
4	2, 3	opeq12i 3163	. . . 4 |- <.(1st` A), (2nd` A)>. = <.U.dom { A}, U.ran { A}>.
5	4	eqeq2i 1894	. . 3 |- (A = <.(1st` A), (2nd` A)>. <-> A = <.U.dom { A}, U.ran { A}>.)
6	2	eleq1i 1960	. . . 4 |- ((1st` A) e. B <-> U.dom { A} e. B)
7	3	eleq1i 1960	. . . 4 |- ((2nd` A) e. C <-> U.ran { A} e. C)
8	6, 7	anbi12i 540	. . 3 |- (((1st` A) e. B /\ (2nd` A) e. C) <-> (U.dom { A} e. B /\ U.ran { A} e. C))
9	5, 8	anbi12i 540	. 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	bitr4i 193	1 |- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {csn 3044  <.cop 3046  U.cuni 3177   X. cxp 3984  dom cdm 3986  ran crn 3987  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  elxp7 5042  eqopi 5043  xpopth 5046  1st2nd 5048  ruclem13 8791  ruclem23 8801  xplmi 9251  xplmi2 9252  bopcnlem2 9260  bopcnlem3 9261  bcthlem4 9280  bcthlem14 9290  vacnlem5 9671  vacnlem6 9672  sspval 9721  eucalg 13755  mulgcdlem2 13757  prj1 14395  issubcat 15193  fnctartar 15284  fnctartar2 15285  difxp 15690  txmet 15925  heiborlem32 15986  heiborlem35 15989  heiborlem36 15990  phtpycolem3 16053  phtpycolem4 16054  phtpycolem5 16055  pcohtpylem3 16082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

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