Theorem op2ndg 4072

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 2478	. . . 4 |- (x = A -> <.x, y>. = <.A, y>.)
2	1	fveq2d 3713	. . 3 |- (x = A -> (2nd` <.x, y>.) = (2nd` <.A, y>.))
3	2	eqeq1d 1475	. 2 |- (x = A -> ((2nd` <.x, y>.) = y <-> (2nd` <.A, y>.) = y))
4		opeq2 2479	. . . 4 |- (y = B -> <.A, y>. = <.A, B>.)
5	4	fveq2d 3713	. . 3 |- (y = B -> (2nd` <.A, y>.) = (2nd` <.A, B>.))
6		id 59	. . 3 |- (y = B -> y = B)
7	5, 6	eqeq12d 1481	. 2 |- (y = B -> ((2nd` <.A, y>.) = y <-> (2nd` <.A, B>.) = B))
8		visset 1804	. . 3 |- x e. V
9		visset 1804	. . 3 |- y e. V
10	8, 9	op2nd 4070	. 2 |- (2nd` <.x, y>.) = y
11	3, 7, 10	vtocl2g 1841	1 |- ((A e. C /\ B e. D) -> (2nd` <.A, B>.) = B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  <.cop 2401  ` cfv 3172  2ndc2nd 4062

This theorem is referenced by:  2ndconst 4081  curry1 4082  eqop 4088  seqzfval 6469  vcoprne 8136

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-2nd 4064

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