Theorem df2nd2 4127

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 1813	. . . 4 |- y e. V
2		visset 1813	. . . . . . 7 |- x e. V
3	2, 1	pm3.2i 285	. . . . . 6 |- (x e. V /\ y e. V)
4	3	biantrur 725	. . . . 5 |- (z = y <-> ((x e. V /\ y e. V) /\ z = y))
5	4	oprabbii 3997	. . . 4 |- {<.<.x, y>., z>. | z = y} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = y)}
6	1, 5	fnoprab2 4122	. . 3 |- {<.<.x, y>., z>. | z = y} Fn (V X. V)
7		fo2nd 4092	. . . . . 6 |- 2nd:V-onto->V
8		fof 3672	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
9	7, 8	ax-mp 7	. . . . 5 |- 2nd:V-->V
10		ffn 3627	. . . . 5 |- (2nd:V-->V -> 2nd Fn V)
11	9, 10	ax-mp 7	. . . 4 |- 2nd Fn V
12		ssv 2081	. . . 4 |- (V X. V) (_ V
13		fnssres 3600	. . . 4 |- ((2nd Fn V /\ (V X. V) (_ V) -> (2nd |` (V X. V)) Fn (V X. V))
14	11, 12, 13	mp2an 697	. . 3 |- (2nd |` (V X. V)) Fn (V X. V)
15		eqfnfv 3797	. . 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 697	. 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 1475	. 2 |- (V X. V) = (V X. V)
18		2nd2val 4096	. . . . . 6 |- ({<.<.x, y>., z>. | z = y}` <.w, v>.) = (2nd` <.w, v>.)
19		visset 1813	. . . . . . . . 9 |- v e. V
20	19	opelxp 3214	. . . . . . . 8 |- (<.w, v>. e. (V X. V) <-> (w e. V /\ v e. V))
21		visset 1813	. . . . . . . 8 |- w e. V
22	20, 21, 19	mpbir2an 730	. . . . . . 7 |- <.w, v>. e. (V X. V)
23		fvres 3734	. . . . . . 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 1498	. . . . 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 1699	. . 3 |- A.w e. V A.v e. V ({<.<.x, y>., z>. | z = y}` <.w, v>.) = ((2nd |` (V X. V))` <.w, v>.)
28		fveq2 3724	. . . . 5 |- (u = <.w, v>. -> ({<.<.x, y>., z>. | z = y}` u) = ({<.<.x, y>., z>. | z = y}` <.w, v>.))
29		fveq2 3724	. . . . 5 |- (u = <.w, v>. -> ((2nd |` (V X. V))` u) = ((2nd |` (V X. V))` <.w, v>.))
30	28, 29	eqeq12d 1489	. . . 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 3218	. . 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 730	1 |- {<.<.x, y>., z>. | z = y} = (2nd |` (V X. V))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  <.cop 2411   X. cxp 3168   |` cres 3172   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  {copab2 3964  2ndc2nd 4078

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-3an 777  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-sbc 1942  df-csb 2002  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-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080

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