Theorem el2xptp0 7212

Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

Assertion

Ref	Expression
el2xptp0	|- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) <-> A = <. X , Y , Z >. ) )

Proof of Theorem el2xptp0

Step	Hyp	Ref	Expression
1		xp1st 7198	. . . . . 6 |- ( A e. ( ( U X. V ) X. W ) -> ( 1st ` A ) e. ( U X. V ) )
2	1	ad2antrl 764	. . . . 5 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( 1st ` A ) e. ( U X. V ) )
3		3simpa 1058	. . . . . . 7 |- ( ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) -> ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y ) )
4	3	adantl 482	. . . . . 6 |- ( ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) -> ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y ) )
5	4	adantl 482	. . . . 5 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y ) )
6		eqopi 7202	. . . . 5 |- ( ( ( 1st ` A ) e. ( U X. V ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y ) ) -> ( 1st ` A ) = <. X , Y >. )
7	2, 5, 6	syl2anc 693	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( 1st ` A ) = <. X , Y >. )
8		simprr3 1111	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( 2nd ` A ) = Z )
9	7, 8	jca 554	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( ( 1st ` A ) = <. X , Y >. /\ ( 2nd ` A ) = Z ) )
10		df-ot 4186	. . . . . 6 |- <. X , Y , Z >. = <. <. X , Y >. , Z >.
11	10	eqeq2i 2634	. . . . 5 |- ( A = <. X , Y , Z >. <-> A = <. <. X , Y >. , Z >. )
12		eqop 7208	. . . . 5 |- ( A e. ( ( U X. V ) X. W ) -> ( A = <. <. X , Y >. , Z >. <-> ( ( 1st ` A ) = <. X , Y >. /\ ( 2nd ` A ) = Z ) ) )
13	11, 12	syl5bb 272	. . . 4 |- ( A e. ( ( U X. V ) X. W ) -> ( A = <. X , Y , Z >. <-> ( ( 1st ` A ) = <. X , Y >. /\ ( 2nd ` A ) = Z ) ) )
14	13	ad2antrl 764	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> ( A = <. X , Y , Z >. <-> ( ( 1st ` A ) = <. X , Y >. /\ ( 2nd ` A ) = Z ) ) )
15	9, 14	mpbird 247	. 2 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) ) -> A = <. X , Y , Z >. )
16		opelxpi 5148	. . . . . . . 8 |- ( ( X e. U /\ Y e. V ) -> <. X , Y >. e. ( U X. V ) )
17	16	3adant3 1081	. . . . . . 7 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> <. X , Y >. e. ( U X. V ) )
18		simp3 1063	. . . . . . 7 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> Z e. W )
19		opelxp 5146	. . . . . . 7 |- ( <. <. X , Y >. , Z >. e. ( ( U X. V ) X. W ) <-> ( <. X , Y >. e. ( U X. V ) /\ Z e. W ) )
20	17, 18, 19	sylanbrc 698	. . . . . 6 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> <. <. X , Y >. , Z >. e. ( ( U X. V ) X. W ) )
21	10, 20	syl5eqel 2705	. . . . 5 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> <. X , Y , Z >. e. ( ( U X. V ) X. W ) )
22	21	adantr 481	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> <. X , Y , Z >. e. ( ( U X. V ) X. W ) )
23		eleq1 2689	. . . . 5 |- ( A = <. X , Y , Z >. -> ( A e. ( ( U X. V ) X. W ) <-> <. X , Y , Z >. e. ( ( U X. V ) X. W ) ) )
24	23	adantl 482	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( A e. ( ( U X. V ) X. W ) <-> <. X , Y , Z >. e. ( ( U X. V ) X. W ) ) )
25	22, 24	mpbird 247	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> A e. ( ( U X. V ) X. W ) )
26		fveq2 6191	. . . . . 6 |- ( A = <. X , Y , Z >. -> ( 1st ` A ) = ( 1st ` <. X , Y , Z >. ) )
27	26	fveq2d 6195	. . . . 5 |- ( A = <. X , Y , Z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` ( 1st ` <. X , Y , Z >. ) ) )
28		ot1stg 7182	. . . . 5 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( 1st ` ( 1st ` <. X , Y , Z >. ) ) = X )
29	27, 28	sylan9eqr 2678	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( 1st ` ( 1st ` A ) ) = X )
30	26	fveq2d 6195	. . . . 5 |- ( A = <. X , Y , Z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` ( 1st ` <. X , Y , Z >. ) ) )
31		ot2ndg 7183	. . . . 5 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( 2nd ` ( 1st ` <. X , Y , Z >. ) ) = Y )
32	30, 31	sylan9eqr 2678	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( 2nd ` ( 1st ` A ) ) = Y )
33		fveq2 6191	. . . . 5 |- ( A = <. X , Y , Z >. -> ( 2nd ` A ) = ( 2nd ` <. X , Y , Z >. ) )
34		ot3rdg 7184	. . . . . 6 |- ( Z e. W -> ( 2nd ` <. X , Y , Z >. ) = Z )
35	34	3ad2ant3 1084	. . . . 5 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( 2nd ` <. X , Y , Z >. ) = Z )
36	33, 35	sylan9eqr 2678	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( 2nd ` A ) = Z )
37	29, 32, 36	3jca 1242	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) )
38	25, 37	jca 554	. 2 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ A = <. X , Y , Z >. ) -> ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) )
39	15, 38	impbida 877	1 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) ) <-> A = <. X , Y , Z >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)