Theorem df1st2 3098

Description: An alternate possible definition of the 1st function.

Assertion

Ref	Expression
df1st2	|- {<.<.x, y>., z>. | z = x} = (1st |` (V X. V))

Distinct variable group(s):   x,y,z

Proof of Theorem df1st2

Step	Hyp	Ref	Expression
1		cleqid 1102	. . 3 |- (V X. V) = (V X. V)
2		1st2val 3097	. . . . . . 7 |- ({<.<.x, y>., z>. | z = x}` <.w, v>.) = (1st` <.w, v>.)
3		visset 1350	. . . . . . . . . 10 |- w e. V
4		visset 1350	. . . . . . . . . 10 |- v e. V
5	3, 4	pm3.2i 234	. . . . . . . . 9 |- (w e. V /\ v e. V)
6	4	opelxp 2452	. . . . . . . . 9 |- (<.w, v>. e. (V X. V) <-> (w e. V /\ v e. V))
7	5, 6	mpbir 165	. . . . . . . 8 |- <.w, v>. e. (V X. V)
8		fvres 2840	. . . . . . . 8 |- (<.w, v>. e. (V X. V) -> ((1st |` (V X. V))` <.w, v>.) = (1st` <.w, v>.))
9	7, 8	ax-mp 6	. . . . . . 7 |- ((1st |` (V X. V))` <.w, v>.) = (1st` <.w, v>.)
10	2, 9	eqtr4 1122	. . . . . 6 |- ({<.<.x, y>., z>. | z = x}` <.w, v>.) = ((1st |` (V X. V))` <.w, v>.)
11	10	a1i 7	. . . . 5 |- ((w e. V /\ v e. V) -> ({<.<.x, y>., z>. | z = x}` <.w, v>.) = ((1st |` (V X. V))` <.w, v>.))
12	11	rgen2 1248	. . . 4 |- A.w e. V A.v e. V ({<.<.x, y>., z>. | z = x}` <.w, v>.) = ((1st |` (V X. V))` <.w, v>.)
13		fveq2 2832	. . . . . 6 |- (u = <.w, v>. -> ({<.<.x, y>., z>. | z = x}` u) = ({<.<.x, y>., z>. | z = x}` <.w, v>.))
14		fveq2 2832	. . . . . 6 |- (u = <.w, v>. -> ((1st |` (V X. V))` u) = ((1st |` (V X. V))` <.w, v>.))
15	13, 14	cleq12d 1115	. . . . 5 |- (u = <.w, v>. -> (({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u) <-> ({<.<.x, y>., z>. | z = x}` <.w, v>.) = ((1st |` (V X. V))` <.w, v>.)))
16	15	ralxp 2456	. . . 4 |- (A.u e. (V X. V)({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u) <-> A.w e. V A.v e. V ({<.<.x, y>., z>. | z = x}` <.w, v>.) = ((1st |` (V X. V))` <.w, v>.))
17	12, 16	mpbir 165	. . 3 |- A.u e. (V X. V)({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u)
18	1, 17	pm3.2i 234	. 2 |- ((V X. V) = (V X. V) /\ A.u e. (V X. V)({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u))
19		visset 1350	. . . 4 |- x e. V
20		visset 1350	. . . . . . 7 |- y e. V
21	19, 20	pm3.2i 234	. . . . . 6 |- (x e. V /\ y e. V)
22	21	biantrur 544	. . . . 5 |- (z = x <-> ((x e. V /\ y e. V) /\ z = x))
23	22	bioprabi 3027	. . . 4 |- {<.<.x, y>., z>. | z = x} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = x)}
24	19, 23	fnoprab2 3039	. . 3 |- {<.<.x, y>., z>. | z = x} Fn (V X. V)
25		fo1st 3094	. . . . . 6 |- 1st:V-onto->V
26		fof 2788	. . . . . 6 |- (1st:V-onto->V -> 1st:V-->V)
27	25, 26	ax-mp 6	. . . . 5 |- 1st:V-->V
28		ffn 2752	. . . . 5 |- (1st:V-->V -> 1st Fn V)
29	27, 28	ax-mp 6	. . . 4 |- 1st Fn V
30		ssv 1520	. . . 4 |- (V X. V) (_ V
31		fnssres 2734	. . . 4 |- ((1st Fn V /\ (V X. V) (_ V) -> (1st |` (V X. V)) Fn (V X. V))
32	29, 30, 31	mp2an 520	. . 3 |- (1st |` (V X. V)) Fn (V X. V)
33		cleqfv 2880	. . 3 |- (({<.<.x, y>., z>. | z = x} Fn (V X. V) /\ (1st |` (V X. V)) Fn (V X. V)) -> ({<.<.x, y>., z>. | z = x} = (1st |` (V X. V)) <-> ((V X. V) = (V X. V) /\ A.u e. (V X. V)({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u))))
34	24, 32, 33	mp2an 520	. 2 |- ({<.<.x, y>., z>. | z = x} = (1st |` (V X. V)) <-> ((V X. V) = (V X. V) /\ A.u e. (V X. V)({<.<.x, y>., z>. | z = x}` u) = ((1st |` (V X. V))` u)))
35	18, 34	mpbir 165	1 |- {<.<.x, y>., z>. | z = x} = (1st |` (V X. V))

Syntax hints:   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487  <.cop 1810   X. cxp 2408   |` cres 2412   Fn wfn 2417  -->wf 2418  -onto->wfo 2420  ` cfv 2422  {copab2 3002  1stc1st 3085

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fo 2436  df-fv 2438  df-opr 3003  df-oprab 3004  df-1st 3087

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