Theorem elxp6 4102

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 3453.

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 3453	. 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 4081	. . . . 5 |- (1st` A) = U.dom { A}
3		2ndval 4082	. . . . 5 |- (2nd` A) = U.ran { A}
4	2, 3	opeq12i 2492	. . . 4 |- <.(1st` A), (2nd` A)>. = <.U.dom { A}, U.ran { A}>.
5	4	eqeq2i 1485	. . 3 |- (A = <.(1st` A), (2nd` A)>. <-> A = <.U.dom { A}, U.ran { A}>.)
6	2	eleq1i 1537	. . . 4 |- ((1st` A) e. B <-> U.dom { A} e. B)
7	3	eleq1i 1537	. . . 4 |- ((2nd` A) e. C <-> U.ran { A} e. C)
8	6, 7	anbi12i 482	. . 3 |- (((1st` A) e. B /\ (2nd` A) e. C) <-> (U.dom { A} e. B /\ U.ran { A} e. C))
9	5, 8	anbi12i 482	. 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 956   e. wcel 958  {csn 2409  <.cop 2411  U.cuni 2503   X. cxp 3168  dom cdm 3170  ran crn 3171  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  elxp7 4103  eqop 4104  xpopth 4106  1st2nd 4108  ruclem13 7522  ruclem23 7532  xplmi 7973  xplmi2 7974  bopcnlem2 7982  bopcnlem3 7983  bcthlem4 8002  bcthlem14 8012  sspval 8382

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080

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