Theorem fconstg 3659

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		feq1 3620	. . . 4 |- ((A X. {x}) = (A X. {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{x}))
2		feq3 3622	. . . 4 |- ({x} = {B} -> ((A X. {B}):A-->{x} <-> (A X. {B}):A-->{B}))
3	1, 2	sylan9bb 540	. . 3 |- (((A X. {x}) = (A X. {B}) /\ {x} = {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{B}))
4		sneq 2417	. . . 4 |- (x = B -> {x} = {B})
5		xpeq2 3201	. . . 4 |- ({x} = {B} -> (A X. {x}) = (A X. {B}))
6	4, 5	syl 10	. . 3 |- (x = B -> (A X. {x}) = (A X. {B}))
7	3, 6, 4	sylanc 471	. 2 |- (x = B -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{B}))
8		visset 1813	. . 3 |- x e. V
9	8	fconst 3658	. 2 |- (A X. {x}):A-->{x}
10	7, 9	vtoclg 1847	1 |- (B e. C -> (A X. {B}):A-->{B})

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {csn 2409   X. cxp 3168  -->wf 3178

This theorem is referenced by:  fvconst2g 3844  fconst2g 3845  exp1t 6573  expp1t 6574  lmconst 7934  opr1cn 7978  opr2cn 7979

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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