Theorem fconst2g 3830

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 3824	. . . . . . . 8 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
2	1	adantlr 393	. . . . . . 7 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = B)
3		fvconst2g 3829	. . . . . . . 8 |- ((B e. C /\ x e. A) -> ((A X. {B})` x) = B)
4	3	adantll 392	. . . . . . 7 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> ((A X. {B})` x) = B)
5	2, 4	eqtr4d 1502	. . . . . 6 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = ((A X. {B})` x))
6	5	r19.21aiva 1706	. . . . 5 |- ((F:A-->{B} /\ B e. C) -> A.x e. A (F` x) = ((A X. {B})` x))
7		eqid 1468	. . . . 5 |- A = A
8	6, 7	jctil 292	. . . 4 |- ((F:A-->{B} /\ B e. C) -> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x)))
9		eqfnfv 3782	. . . . 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))))
10		ffn 3613	. . . . 5 |- (F:A-->{B} -> F Fn A)
11		fconstg 3644	. . . . . 6 |- (B e. C -> (A X. {B}):A-->{B})
12		ffn 3613	. . . . . 6 |- ((A X. {B}):A-->{B} -> (A X. {B}) Fn A)
13	11, 12	syl 10	. . . . 5 |- (B e. C -> (A X. {B}) Fn A)
14	9, 10, 13	syl2an 454	. . . 4 |- ((F:A-->{B} /\ B e. C) -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
15	8, 14	mpbird 196	. . 3 |- ((F:A-->{B} /\ B e. C) -> F = (A X. {B}))
16	15	expcom 374	. 2 |- (B e. C -> (F:A-->{B} -> F = (A X. {B})))
17		feq1 3606	. . 3 |- (F = (A X. {B}) -> (F:A-->{B} <-> (A X. {B}):A-->{B}))
18	17, 11	syl5cbir 211	. 2 |- (B e. C -> (F = (A X. {B}) -> F:A-->{B}))
19	16, 18	impbid 514	1 |- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  {csn 2399   X. cxp 3158   Fn wfn 3167  -->wf 3168  ` cfv 3172

This theorem is referenced by:  fconst2 3832  fconst5 3833

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188

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