Theorem xpopth 4096

Description: An ordered pair theorem for members of cross products.

Assertion

Ref	Expression
xpopth	|- ((A e. (C X. D) /\ B e. (R X. S)) -> (((1st` A) = (1st` B) /\ (2nd` A) = (2nd` B)) <-> A = B))

Proof of Theorem xpopth

Step	Hyp	Ref	Expression
1		elxp6 4092	. . . 4 |- (A e. (C X. D) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. C /\ (2nd` A) e. D)))
2	1	pm3.26bi 322	. . 3 |- (A e. (C X. D) -> A = <.(1st` A), (2nd` A)>.)
3		elxp6 4092	. . . 4 |- (B e. (R X. S) <-> (B = <.(1st` B), (2nd` B)>. /\ ((1st` B) e. R /\ (2nd` B) e. S)))
4	3	pm3.26bi 322	. . 3 |- (B e. (R X. S) -> B = <.(1st` B), (2nd` B)>.)
5	2, 4	eqeqan12d 1487	. 2 |- ((A e. (C X. D) /\ B e. (R X. S)) -> (A = B <-> <.(1st` A), (2nd` A)>. = <.(1st` B), (2nd` B)>.))
6		fvex 3723	. . 3 |- (1st` A) e. V
7		fvex 3723	. . 3 |- (2nd` A) e. V
8		fvex 3723	. . 3 |- (2nd` B) e. V
9	6, 7, 8	opth 2782	. 2 |- (<.(1st` A), (2nd` A)>. = <.(1st` B), (2nd` B)>. <-> ((1st` A) = (1st` B) /\ (2nd` A) = (2nd` B)))
10	5, 9	syl6rbb 536	1 |- ((A e. (C X. D) /\ B e. (R X. S)) -> (((1st` A) = (1st` B) /\ (2nd` A) = (2nd` B)) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  <.cop 2407   X. cxp 3163  ` cfv 3177  1stc1st 4067  2ndc2nd 4068

This theorem is referenced by:  metxp 7786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-1st 4069  df-2nd 4070

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