Theorem fconstOLD 4603

Description: A cross product with a singleton is a constant function.

Hypothesis

Ref	Expression
fconst.1	|- B e. _V

Assertion

Ref	Expression
fconstOLD	|- (A X. {B}):A-->{B}

Proof of Theorem fconstOLD

Step	Hyp	Ref	Expression
1		f0 4600	. . 3 |- (/):(/)-->{B}
2		xpeq1 4016	. . . . . 6 |- (A = (/) -> (A X. {B}) = ((/) X. {B}))
3		xp0r 4065	. . . . . 6 |- ((/) X. {B}) = (/)
4	2, 3	syl6eq 1944	. . . . 5 |- (A = (/) -> (A X. {B}) = (/))
5	4	feq1d 4556	. . . 4 |- (A = (/) -> ((A X. {B}):A-->{B} <-> (/):A-->{B}))
6		feq2 4552	. . . 4 |- (A = (/) -> ((/):A-->{B} <-> (/):(/)-->{B}))
7	5, 6	bitrd 587	. . 3 |- (A = (/) -> ((A X. {B}):A-->{B} <-> (/):(/)-->{B}))
8	1, 7	mpbiri 211	. 2 |- (A = (/) -> (A X. {B}):A-->{B})
9		rnxp 4342	. . . . 5 |- (A =/= (/) -> ran ( A X. {B}) = {B})
10		eqimss 2665	. . . . 5 |- (ran ( A X. {B}) = {B} -> ran ( A X. {B}) C_ {B})
11	9, 10	syl 12	. . . 4 |- (A =/= (/) -> ran ( A X. {B}) C_ {B})
12		df-fn 4009	. . . . 5 |- ((A X. {B}) Fn A <-> (Fun (A X. {B}) /\ dom ( A X. {B}) = A))
13		dffun6 4436	. . . . . 6 |- (Fun (A X. {B}) <-> (Rel (A X. {B}) /\ A.xE*y x(A X. {B})y))
14		relxp 4088	. . . . . 6 |- Rel (A X. {B})
15		moeq 2431	. . . . . . . . 9 |- E*y y = B
16	15	moani 1820	. . . . . . . 8 |- E*y(x e. A /\ y = B)
17		visset 2295	. . . . . . . . . . 11 |- y e. _V
18	17	brxp 4038	. . . . . . . . . 10 |- (x(A X. {B})y <-> (x e. A /\ y e. {B}))
19		elsn 3058	. . . . . . . . . . 11 |- (y e. {B} <-> y = B)
20	19	anbi2i 538	. . . . . . . . . 10 |- ((x e. A /\ y e. {B}) <-> (x e. A /\ y = B))
21	18, 20	bitri 190	. . . . . . . . 9 |- (x(A X. {B})y <-> (x e. A /\ y = B))
22	21	mobii 1801	. . . . . . . 8 |- (E*y x(A X. {B})y <-> E*y(x e. A /\ y = B))
23	16, 22	mpbir 207	. . . . . . 7 |- E*y x(A X. {B})y
24	23	ax-gen 1305	. . . . . 6 |- A.xE*y x(A X. {B})y
25	13, 14, 24	mpbir2an 800	. . . . 5 |- Fun (A X. {B})
26		fconst.1	. . . . . . 7 |- B e. _V
27	26	snnz 3119	. . . . . 6 |- {B} =/= (/)
28		dmxp 4177	. . . . . 6 |- ({B} =/= (/) -> dom ( A X. {B}) = A)
29	27, 28	ax-mp 7	. . . . 5 |- dom ( A X. {B}) = A
30	12, 25, 29	mpbir2an 800	. . . 4 |- (A X. {B}) Fn A
31	11, 30	jctil 316	. . 3 |- (A =/= (/) -> ((A X. {B}) Fn A /\ ran ( A X. {B}) C_ {B}))
32		df-f 4010	. . 3 |- ((A X. {B}):A-->{B} <-> ((A X. {B}) Fn A /\ ran ( A X. {B}) C_ {B}))
33	31, 32	sylibr 217	. 2 |- (A =/= (/) -> (A X. {B}):A-->{B})
34	8, 33	pm2.61ine 2089	1 |- (A X. {B}):A-->{B}

Syntax hints:   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  -->wf 3994

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  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-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-fun 4008  df-fn 4009  df-f 4010

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