Theorem fconstg 4604

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

Assertion

Ref	Expression
fconstg	|- (B e. C -> (A X. {B}):A-->{B})

Proof of Theorem fconstg

Step	Hyp	Ref	Expression
1		sneq 3054	. . . 4 |- (x = B -> {x} = {B})
2		xpeq2 4017	. . . 4 |- ({x} = {B} -> (A X. {x}) = (A X. {B}))
3	1, 2	syl 12	. . 3 |- (x = B -> (A X. {x}) = (A X. {B}))
4		feq1 4551	. . . 4 |- ((A X. {x}) = (A X. {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{x}))
5		feq3 4553	. . . 4 |- ({x} = {B} -> ((A X. {B}):A-->{x} <-> (A X. {B}):A-->{B}))
6	4, 5	sylan9bb 599	. . 3 |- (((A X. {x}) = (A X. {B}) /\ {x} = {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{B}))
7	3, 1, 6	syl11anc 524	. 2 |- (x = B -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{B}))
8		visset 2295	. . 3 |- x e. _V
9	8	fconst 4602	. 2 |- (A X. {x}):A-->{x}
10	7, 9	vtoclg 2346	1 |- (B e. C -> (A X. {B}):A-->{B})

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {csn 3044   X. cxp 3984  -->wf 3994

This theorem is referenced by:  fvconst2g 4820  fconst2g 4821  exp1 7816  expp1 7817  lmconst 9212  opr1cn 9256  opr2cn 9257  gx1 9385  gxnn0suc 9387  algrf 13739  constcncf 15882  tlmconst 15907  reheibor 16025  pcorev 16087  pi1gp 16095

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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