Theorem op2ndg 5029

Description: Extract the second member of an ordered pair.

Assertion

Ref	Expression
op2ndg	|- ((A e. C /\ B e. D) -> (2nd` <.A, B>.) = B)

Proof of Theorem op2ndg

Step	Hyp	Ref	Expression
1		opeq1 3158	. . . 4 |- (x = A -> <.x, y>. = <.A, y>.)
2	1	fveq2d 4685	. . 3 |- (x = A -> (2nd` <.x, y>.) = (2nd` <.A, y>.))
3	2	eqeq1d 1892	. 2 |- (x = A -> ((2nd` <.x, y>.) = y <-> (2nd` <.A, y>.) = y))
4		opeq2 3159	. . . 4 |- (y = B -> <.A, y>. = <.A, B>.)
5	4	fveq2d 4685	. . 3 |- (y = B -> (2nd` <.A, y>.) = (2nd` <.A, B>.))
6		id 73	. . 3 |- (y = B -> y = B)
7	5, 6	eqeq12d 1899	. 2 |- (y = B -> ((2nd` <.A, y>.) = y <-> (2nd` <.A, B>.) = B))
8		visset 2295	. . 3 |- x e. _V
9		visset 2295	. . 3 |- y e. _V
10	8, 9	op2nd 5027	. 2 |- (2nd` <.x, y>.) = y
11	3, 7, 10	vtocl2g 2349	1 |- ((A e. C /\ B e. D) -> (2nd` <.A, B>.) = B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  <.cop 3046  ` cfv 3998  2ndc2nd 5019

This theorem is referenced by:  eqop 5044  2ndconst 5071  curry1 5075  seqzfval 7780  gapmlem 9461  vcoprne 9530  eucalgval2 13750  eucalg 13755

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-2nd 5021

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