Theorem fconstg 3654

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 3616	. . . 4 |- ((A X. {x}) = (A X. {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{x}))
2		feq3 3618	. . . 4 |- ({x} = {B} -> ((A X. {B}):A-->{x} <-> (A X. {B}):A-->{B}))
3	1, 2	sylan9bb 539	. . 3 |- (((A X. {x}) = (A X. {B}) /\ {x} = {B}) -> ((A X. {x}):A-->{x} <-> (A X. {B}):A-->{B}))
4		sneq 2414	. . . 4 |- (x = B -> {x} = {B})
5		xpeq2 3197	. . . 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 1810	. . 3 |- x e. V
9	8	fconst 3653	. 2 |- (A X. {x}):A-->{x}
10	7, 9	vtoclg 1844	1 |- (B e. C -> (A X. {B}):A-->{B})

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  {csn 2406   X. cxp 3164  -->wf 3174

This theorem is referenced by:  fvconst2g 3839  fconst2g 3840  exp1t 6518  expp1t 6519  lmconst 7896  opr1cn 7940  opr2cn 7941

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189  df-f 3190

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