Theorem eqop 4104

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 1534	. . . . 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 3205	. . . 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 3724	. . . . 5 |- (A = <.B, C>. -> (1st` A) = (1st` <.B, C>.))
6		op1stg 4087	. . . . 5 |- (B e. V -> (1st` <.B, C>.) = B)
7	5, 6	sylan9eqr 1529	. . . 4 |- ((B e. V /\ A = <.B, C>.) -> (1st` A) = B)
8		fveq2 3724	. . . . 5 |- (A = <.B, C>. -> (2nd` A) = (2nd` <.B, C>.))
9		eqop.1	. . . . . 6 |- C e. V
10		op2ndg 4088	. . . . . 6 |- ((B e. V /\ C e. V) -> (2nd` <.B, C>.) = C)
11	9, 10	mpan2 696	. . . . 5 |- (B e. V -> (2nd` <.B, C>.) = C)
12	8, 11	sylan9eqr 1529	. . . 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 4102	. . . 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 2489	. . 3 |- (((1st` A) = B /\ (2nd` A) = C) -> <.(1st` A), (2nd` A)>. = <.B, C>.)
18	16, 17	sylan9eq 1527	. 2 |- ((A e. (V X. V) /\ ((1st` A) = B /\ (2nd` A) = C)) -> A = <.B, C>.)
19	14, 18	impbida 519	1 |- (A e. (V X. V) -> (A = <.B, C>. <-> ((1st` A) = B /\ (2nd` A) = C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411   X. cxp 3168  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  dfoprab5 4115

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080

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