Theorem eqop 5044

Description: Two ways to express equality with an ordered pair.

Hypothesis

Ref	Expression
eqop.1	|- C e. _V

Assertion

Ref	Expression
eqop	|- (A e. (_V X. _V) -> (A = <.B, C>. <-> ((1st` A) = B /\ (2nd` A) = C)))

Proof of Theorem eqop

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (A = <.B, C>. -> (A e. (_V X. _V) <-> <.B, C>. e. (_V X. _V)))
2	1	biimpac 462	. . . 4 |- ((A e. (_V X. _V) /\ A = <.B, C>.) -> <.B, C>. e. (_V X. _V))
3		opelxp1 4026	. . . 4 |- (<.B, C>. e. (_V X. _V) -> B e. _V)
4	2, 3	syl 12	. . 3 |- ((A e. (_V X. _V) /\ A = <.B, C>.) -> B e. _V)
5		fveq2 4681	. . . . 5 |- (A = <.B, C>. -> (1st` A) = (1st` <.B, C>.))
6		op1stg 5028	. . . . 5 |- (B e. _V -> (1st` <.B, C>.) = B)
7	5, 6	sylan9eqr 1951	. . . 4 |- ((B e. _V /\ A = <.B, C>.) -> (1st` A) = B)
8		fveq2 4681	. . . . 5 |- (A = <.B, C>. -> (2nd` A) = (2nd` <.B, C>.))
9		eqop.1	. . . . . 6 |- C e. _V
10		op2ndg 5029	. . . . . 6 |- ((B e. _V /\ C e. _V) -> (2nd` <.B, C>.) = C)
11	9, 10	mpan2 760	. . . . 5 |- (B e. _V -> (2nd` <.B, C>.) = C)
12	8, 11	sylan9eqr 1951	. . . 4 |- ((B e. _V /\ A = <.B, C>.) -> (2nd` A) = C)
13	7, 12	jca 310	. . 3 |- ((B e. _V /\ A = <.B, C>.) -> ((1st` A) = B /\ (2nd` A) = C))
14	4, 13	sylancom 531	. 2 |- ((A e. (_V X. _V) /\ A = <.B, C>.) -> ((1st` A) = B /\ (2nd` A) = C))
15		eqopi 5043	. 2 |- ((A e. (_V X. _V) /\ ((1st` A) = B /\ (2nd` A) = C)) -> A = <.B, C>.)
16	14, 15	impbida 577	1 |- (A e. (_V X. _V) -> (A = <.B, C>. <-> ((1st` A) = B /\ (2nd` A) = C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046   X. cxp 3984  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

Copyright terms: Public domain