Theorem df2nd2 5069

Description: An alternate possible definition of the 2nd function.

Assertion

Ref	Expression
df2nd2	|- {<.<.x, y>., z>. | z = y} = (2nd |` (_V X. _V))

Distinct variable group:   x,y,z

Proof of Theorem df2nd2

Step	Hyp	Ref	Expression
1		resopab 4252	. 2 |- ({<.w, z>. | z = (2nd` w)} |` (_V X. _V)) = {<.w, z>. | (w e. (_V X. _V) /\ z = (2nd` w))}
2		fo2nd 5033	. . . . . 6 |- 2nd:_V-onto->_V
3		fofn 4619	. . . . . 6 |- (2nd:_V-onto->_V -> 2nd Fn _V)
4	2, 3	ax-mp 7	. . . . 5 |- 2nd Fn _V
5		dffn5 4717	. . . . 5 |- (2nd Fn _V <-> 2nd = {<.w, z>. | (w e. _V /\ z = (2nd` w))})
6	4, 5	mpbi 206	. . . 4 |- 2nd = {<.w, z>. | (w e. _V /\ z = (2nd` w))}
7		visset 2295	. . . . . 6 |- w e. _V
8	7	biantrur 794	. . . . 5 |- (z = (2nd` w) <-> (w e. _V /\ z = (2nd` w)))
9	8	opabbii 3402	. . . 4 |- {<.w, z>. | z = (2nd` w)} = {<.w, z>. | (w e. _V /\ z = (2nd` w))}
10	6, 9	eqtr4i 1911	. . 3 |- 2nd = {<.w, z>. | z = (2nd` w)}
11		reseq1 4218	. . 3 |- (2nd = {<.w, z>. | z = (2nd` w)} -> (2nd |` (_V X. _V)) = ({<.w, z>. | z = (2nd` w)} |` (_V X. _V)))
12	10, 11	ax-mp 7	. 2 |- (2nd |` (_V X. _V)) = ({<.w, z>. | z = (2nd` w)} |` (_V X. _V))
13		fveq2 4681	. . . . 5 |- (<.x, y>. = w -> (2nd` <.x, y>.) = (2nd` w))
14		visset 2295	. . . . . 6 |- x e. _V
15		visset 2295	. . . . . 6 |- y e. _V
16	14, 15	op2nd 5027	. . . . 5 |- (2nd` <.x, y>.) = y
17	13, 16	syl5eqr 1942	. . . 4 |- (<.x, y>. = w -> y = (2nd` w))
18	17	eqeq2d 1895	. . 3 |- (<.x, y>. = w -> (z = y <-> z = (2nd` w)))
19	18	dfoprab3s 5055	. 2 |- {<.<.x, y>., z>. | z = y} = {<.w, z>. | (w e. (_V X. _V) /\ z = (2nd` w))}
20	1, 12, 19	3eqtr4ri 1923	1 |- {<.<.x, y>., z>. | z = y} = (2nd |` (_V X. _V))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  {copab 3395   X. cxp 3984   |` cres 3988   Fn wfn 3993  -onto->wfo 3996  ` cfv 3998  {copab2 4885  2ndc2nd 5019

This theorem is referenced by:  fparlem4 5084  brtxp 14067

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-3an 860  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-sbc 2454  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-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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