Theorem oteqimp 30300

Description: The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

Assertion

Ref	Expression
oteqimp	|- ( T = <. A , B , C >. -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = B /\ ( 2nd ` T ) = C ) ) )

Proof of Theorem oteqimp

Step	Hyp	Ref	Expression
1		ot1stg 6704	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A )
2		ot2ndg 6705	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B )
3		ot3rdg 6706	. . . 4 |- ( C e. Z -> ( 2nd ` <. A , B , C >. ) = C )
4	3	3ad2ant3 1011	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( 2nd ` <. A , B , C >. ) = C )
5	1, 2, 4	3jca 1168	. 2 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( 1st ` ( 1st ` <. A , B , C >. ) ) = A /\ ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B /\ ( 2nd ` <. A , B , C >. ) = C ) )
6		fveq2 5802	. . . . 5 |- ( T = <. A , B , C >. -> ( 1st ` T ) = ( 1st ` <. A , B , C >. ) )
7	6	fveq2d 5806	. . . 4 |- ( T = <. A , B , C >. -> ( 1st ` ( 1st ` T ) ) = ( 1st ` ( 1st ` <. A , B , C >. ) ) )
8	7	eqeq1d 2456	. . 3 |- ( T = <. A , B , C >. -> ( ( 1st ` ( 1st ` T ) ) = A <-> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A ) )
9	6	fveq2d 5806	. . . 4 |- ( T = <. A , B , C >. -> ( 2nd ` ( 1st ` T ) ) = ( 2nd ` ( 1st ` <. A , B , C >. ) ) )
10	9	eqeq1d 2456	. . 3 |- ( T = <. A , B , C >. -> ( ( 2nd ` ( 1st ` T ) ) = B <-> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B ) )
11		fveq2 5802	. . . 4 |- ( T = <. A , B , C >. -> ( 2nd ` T ) = ( 2nd ` <. A , B , C >. ) )
12	11	eqeq1d 2456	. . 3 |- ( T = <. A , B , C >. -> ( ( 2nd ` T ) = C <-> ( 2nd ` <. A , B , C >. ) = C ) )
13	8, 10, 12	3anbi123d 1290	. 2 |- ( T = <. A , B , C >. -> ( ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = B /\ ( 2nd ` T ) = C ) <-> ( ( 1st ` ( 1st ` <. A , B , C >. ) ) = A /\ ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B /\ ( 2nd ` <. A , B , C >. ) = C ) ) )
14	5, 13	syl5ibr 221	1 |- ( T = <. A , B , C >. -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = B /\ ( 2nd ` T ) = C ) ) )

Syntax hints:   -> wi 4   /\ w3a 965   = wceq 1370   e. wcel 1758   <.cotp 3996   ` cfv 5529   1stc1st 6688   2ndc2nd 6689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691

This theorem is referenced by:  el2wlkonot  30559  el2spthonot  30560  el2spthonot0  30561