Theorem fconst2g 4821

Description: A constant function expressed as a cross product.

Assertion

Ref	Expression
fconst2g	|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))

Proof of Theorem fconst2g

Step	Hyp	Ref	Expression
1		fvconst 4814	. . . . . . 7 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
2	1	adantlr 429	. . . . . 6 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = B)
3		fvconst2g 4820	. . . . . . 7 |- ((B e. C /\ x e. A) -> ((A X. {B})` x) = B)
4	3	adantll 428	. . . . . 6 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> ((A X. {B})` x) = B)
5	2, 4	eqtr4d 1928	. . . . 5 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = ((A X. {B})` x))
6	5	r19.21aiva 2176	. . . 4 |- ((F:A-->{B} /\ B e. C) -> A.x e. A (F` x) = ((A X. {B})` x))
7		eqfnfv2 4767	. . . . 5 |- ((F Fn A /\ (A X. {B}) Fn A) -> (F = (A X. {B}) <-> A.x e. A (F` x) = ((A X. {B})` x)))
8		ffn 4562	. . . . 5 |- (F:A-->{B} -> F Fn A)
9		fconstg 4604	. . . . . 6 |- (B e. C -> (A X. {B}):A-->{B})
10		ffn 4562	. . . . . 6 |- ((A X. {B}):A-->{B} -> (A X. {B}) Fn A)
11	9, 10	syl 12	. . . . 5 |- (B e. C -> (A X. {B}) Fn A)
12	7, 8, 11	syl2an 503	. . . 4 |- ((F:A-->{B} /\ B e. C) -> (F = (A X. {B}) <-> A.x e. A (F` x) = ((A X. {B})` x)))
13	6, 12	mpbird 213	. . 3 |- ((F:A-->{B} /\ B e. C) -> F = (A X. {B}))
14	13	expcom 403	. 2 |- (B e. C -> (F:A-->{B} -> F = (A X. {B})))
15		feq1 4551	. . 3 |- (F = (A X. {B}) -> (F:A-->{B} <-> (A X. {B}):A-->{B}))
16	15, 9	syl5cbir 228	. 2 |- (B e. C -> (F = (A X. {B}) -> F:A-->{B}))
17	14, 16	impbid 574	1 |- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {csn 3044   X. cxp 3984   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  fconst2 4823  fconst5 4824

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-13 1311  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  ax-un 3790

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-rex 2110  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-uni 3178  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-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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