Theorem eqop 4094

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 1531	. . . . 5 |- (A = <.B, C>. -> (A e. (V X. V) <-> <.B, C>. e. (V X. V)))
2	1	biimpac 418	. . . 4 |- ((A e. (V X. V) /\ A = <.B, C>.) -> <.B, C>. e. (V X. V))
3		opelxp1 3200	. . . 4 |- (<.B, C>. e. (V X. V) -> B e. V)
4	2, 3	syl 10	. . 3 |- ((A e. (V X. V) /\ A = <.B, C>.) -> B e. V)
5		fveq2 3715	. . . . 5 |- (A = <.B, C>. -> (1st` A) = (1st` <.B, C>.))
6		op1stg 4077	. . . . 5 |- (B e. V -> (1st` <.B, C>.) = B)
7	5, 6	sylan9eqr 1526	. . . 4 |- ((B e. V /\ A = <.B, C>.) -> (1st` A) = B)
8		fveq2 3715	. . . . 5 |- (A = <.B, C>. -> (2nd` A) = (2nd` <.B, C>.))
9		eqop.1	. . . . . 6 |- C e. V
10		op2ndg 4078	. . . . . 6 |- ((B e. V /\ C e. V) -> (2nd` <.B, C>.) = C)
11	9, 10	mpan2 695	. . . . 5 |- (B e. V -> (2nd` <.B, C>.) = C)
12	8, 11	sylan9eqr 1526	. . . 4 |- ((B e. V /\ A = <.B, C>.) -> (2nd` A) = C)
13	7, 12	jca 288	. . 3 |- ((B e. V /\ A = <.B, C>.) -> ((1st` A) = B /\ (2nd` A) = C))
14	4, 13	sylancom 475	. 2 |- ((A e. (V X. V) /\ A = <.B, C>.) -> ((1st` A) = B /\ (2nd` A) = C))
15		elxp6 4092	. . . 4 |- (A e. (V X. V) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. V /\ (2nd` A) e. V)))
16	15	pm3.26bi 322	. . 3 |- (A e. (V X. V) -> A = <.(1st` A), (2nd` A)>.)
17		opeq12 2485	. . 3 |- (((1st` A) = B /\ (2nd` A) = C) -> <.(1st` A), (2nd` A)>. = <.B, C>.)
18	16, 17	sylan9eq 1524	. 2 |- ((A e. (V X. V) /\ ((1st` A) = B /\ (2nd` A) = C)) -> A = <.B, C>.)
19	14, 18	impbida 518	1 |- (A e. (V X. V) -> (A = <.B, C>. <-> ((1st` A) = B /\ (2nd` A) = C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  <.cop 2407   X. cxp 3163  ` cfv 3177  1stc1st 4067  2ndc2nd 4068

This theorem is referenced by:  dfoprab5 4105

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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