Theorem elxp7 4109

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 3459.

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 4108	. 2 |- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))
2		elxp6 4108	. . . 4 |- (A e. (V X. V) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. V /\ (2nd` A) e. V)))
3		fvex 3738	. . . . 5 |- (1st` A) e. V
4		fvex 3738	. . . . 5 |- (2nd` A) e. V
5	3, 4	pm3.2i 285	. . . 4 |- ((1st` A) e. V /\ (2nd` A) e. V)
6	2, 5	mpbiran2 731	. . 3 |- (A e. (V X. V) <-> A = <.(1st` A), (2nd` A)>.)
7	6	anbi1i 483	. 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 958   e. wcel 960  Vcvv 1814  <.cop 2415   X. cxp 3174  ` cfv 3188  1stc1st 4083  2ndc2nd 4084

This theorem is referenced by:  xp2 4111  unielxp 4113  dfoprab5 4121  foprab2 4125  metxplem1 7823  metxplem2 7824  metxplem4 7830  metxp 7831  xplmi 7970

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-1st 4085  df-2nd 4086

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