Theorem df2nd2 4133

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		visset 1816	. . . 4 |- y e. V
2		visset 1816	. . . . . . 7 |- x e. V
3	2, 1	pm3.2i 285	. . . . . 6 |- (x e. V /\ y e. V)
4	3	biantrur 727	. . . . 5 |- (z = y <-> ((x e. V /\ y e. V) /\ z = y))
5	4	oprabbii 4003	. . . 4 |- {<.<.x, y>., z>. | z = y} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = y)}
6	1, 5	fnoprab2 4128	. . 3 |- {<.<.x, y>., z>. | z = y} Fn (V X. V)
7		fo2nd 4098	. . . . . 6 |- 2nd:V-onto->V
8		fof 3678	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
9	7, 8	ax-mp 7	. . . . 5 |- 2nd:V-->V
10		ffn 3633	. . . . 5 |- (2nd:V-->V -> 2nd Fn V)
11	9, 10	ax-mp 7	. . . 4 |- 2nd Fn V
12		ssv 2084	. . . 4 |- (V X. V) (_ V
13		fnssres 3606	. . . 4 |- ((2nd Fn V /\ (V X. V) (_ V) -> (2nd |` (V X. V)) Fn (V X. V))
14	11, 12, 13	mp2an 699	. . 3 |- (2nd |` (V X. V)) Fn (V X. V)
15		eqfnfv 3803	. . 3 |- (({<.<.x, y>., z>. | z = y} Fn (V X. V) /\ (2nd |` (V X. V)) Fn (V X. V)) -> ({<.<.x, y>., z>. | z = y} = (2nd |` (V X. V)) <-> ((V X. V) = (V X. V) /\ A.u e. (V X. V)({<.<.x, y>., z>. | z = y}` u) = ((2nd |` (V X. V))` u))))
16	6, 14, 15	mp2an 699	. 2 |- ({<.<.x, y>., z>. | z = y} = (2nd |` (V X. V)) <-> ((V X. V) = (V X. V) /\ A.u e. (V X. V)({<.<.x, y>., z>. | z = y}` u) = ((2nd |` (V X. V))` u)))
17		eqid 1478	. 2 |- (V X. V) = (V X. V)
18		2nd2val 4102	. . . . . 6 |- ({<.<.x, y>., z>. | z = y}` <.w, v>.) = (2nd` <.w, v>.)
19		visset 1816	. . . . . . . . 9 |- v e. V
20	19	opelxp 3220	. . . . . . . 8 |- (<.w, v>. e. (V X. V) <-> (w e. V /\ v e. V))
21		visset 1816	. . . . . . . 8 |- w e. V
22	20, 21, 19	mpbir2an 732	. . . . . . 7 |- <.w, v>. e. (V X. V)
23		fvres 3740	. . . . . . 7 |- (<.w, v>. e. (V X. V) -> ((2nd |` (V X. V))` <.w, v>.) = (2nd` <.w, v>.))
24	22, 23	ax-mp 7	. . . . . 6 |- ((2nd |` (V X. V))` <.w, v>.) = (2nd` <.w, v>.)
25	18, 24	eqtr4 1501	. . . . 5 |- ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.)
26	25	a1i 8	. . . 4 |- ((w e. V /\ v e. V) -> ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.))
27	26	rgen2a 1702	. . 3 |- A.w e. V A.v e. V ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.)
28		fveq2 3730	. . . . 5 |- (u = <.w, v>. -> ({<.<.x, y>., z>. | z = y}` u) = ({<.<.x, y>., z>. | z = y}` <.w, v>.))
29		fveq2 3730	. . . . 5 |- (u = <.w, v>. -> ((2nd |` (V X. V))` u) = ((2nd |` (V X. V))` <.w, v>.))
30	28, 29	eqeq12d 1492	. . . 4 |- (u = <.w, v>. -> (({<.<.x, y>., z>. | z = y}` u) = ((2nd |` (V X. V))` u) <-> ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.)))
31	30	ralxp 3224	. . 3 |- (A.u e. (V X. V)({<.<.x, y>., z>. | z = y}` u) = ((2nd |` (V X. V))` u) <-> A.w e. V A.v e. V ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.))
32	27, 31	mpbir 190	. 2 |- A.u e. (V X. V)({<.<.x, y>., z>. | z = y}` u) = ((2nd |` (V X. V))` u)
33	16, 17, 32	mpbir2an 732	1 |- {<.<.x, y>., z>. | z = y} = (2nd |` (V X. V))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814   (_ wss 2050  <.cop 2415   X. cxp 3174   |` cres 3178   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  ` cfv 3188  {copab2 3970  2ndc2nd 4084

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086

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