Theorem elxp7 4103

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

Assertion

Ref	Expression
elxp7	|- (A e. (B X. C) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))

Proof of Theorem elxp7

Step	Hyp	Ref	Expression
1		elxp6 4102	. 2 |- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))
2		elxp6 4102	. . . 4 |- (A e. (V X. V) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. V /\ (2nd` A) e. V)))
3		fvex 3732	. . . . 5 |- (1st` A) e. V
4		fvex 3732	. . . . 5 |- (2nd` A) e. V
5	3, 4	pm3.2i 285	. . . 4 |- ((1st` A) e. V /\ (2nd` A) e. V)
6	2, 5	mpbiran2 729	. . 3 |- (A e. (V X. V) <-> A = <.(1st` A), (2nd` A)>.)
7	6	anbi1i 481	. 2 |- ((A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))
8	1, 7	bitr4 176	1 |- (A e. (B X. C) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411   X. cxp 3168  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  xp2 4105  unielxp 4107  dfoprab5 4115  foprab2 4119  metxplem1 7826  metxplem2 7827  metxplem4 7833  metxp 7834  xplmi 7973

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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