Theorem fconst2 2902

Description: A constant function expressed as a cross product. See also fconstfv 2903.

Hypothesis

Ref	Expression
fconst2.1	|- B e. V

Assertion

Ref	Expression
fconst2	|- (F:A-->{B} <-> F = (A X. {B}))

Proof of Theorem fconst2

Step	Hyp	Ref	Expression
1		fvconst 2899	. . . . . . 7 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
2		fconst2.1	. . . . . . . . . 10 |- B e. V
3	2	fconst 2774	. . . . . . . . 9 |- (A X. {B}):A-->{B}
4		fvconst 2899	. . . . . . . . 9 |- (((A X. {B}):A-->{B} /\ x e. A) -> ((A X. {B})` x) = B)
5	3, 4	mpan 518	. . . . . . . 8 |- (x e. A -> ((A X. {B})` x) = B)
6	5	adantl 305	. . . . . . 7 |- ((F:A-->{B} /\ x e. A) -> ((A X. {B})` x) = B)
7	1, 6	eqtr4d 1131	. . . . . 6 |- ((F:A-->{B} /\ x e. A) -> (F` x) = ((A X. {B})` x))
8	7	exp 291	. . . . 5 |- (F:A-->{B} -> (x e. A -> (F` x) = ((A X. {B})` x)))
9	8	r19.21aiv 1259	. . . 4 |- (F:A-->{B} -> A.x e. A (F` x) = ((A X. {B})` x))
10		cleqid 1102	. . . 4 |- A = A
11	9, 10	jctil 240	. . 3 |- (F:A-->{B} -> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x)))
12		ffn 2752	. . . 4 |- (F:A-->{B} -> F Fn A)
13		ffn 2752	. . . . . 6 |- ((A X. {B}):A-->{B} -> (A X. {B}) Fn A)
14	3, 13	ax-mp 6	. . . . 5 |- (A X. {B}) Fn A
15		cleqfv 2880	. . . . 5 |- ((F Fn A /\ (A X. {B}) Fn A) -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
16	14, 15	mpan2 519	. . . 4 |- (F Fn A -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
17	12, 16	syl 12	. . 3 |- (F:A-->{B} -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
18	11, 17	mpbird 171	. 2 |- (F:A-->{B} -> F = (A X. {B}))
19		feq1 2748	. . 3 |- (F = (A X. {B}) -> (F:A-->{B} <-> (A X. {B}):A-->{B}))
20	3, 19	mpbiri 169	. 2 |- (F = (A X. {B}) -> F:A-->{B})
21	18, 20	impbi 139	1 |- (F:A-->{B} <-> F = (A X. {B}))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348  {csn 1808   X. cxp 2408   Fn wfn 2417  -->wf 2418  ` cfv 2422

This theorem is referenced by:  map1 3335  hsn0elch 5155  ho0 5612

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-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-fv 2438

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