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Theorem fconstg 5026
 Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
fconstg (B 𝑉 → (A × {B}):A⟶{B})

Proof of Theorem fconstg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . 4 (x = B → {x} = {B})
21xpeq2d 4312 . . 3 (x = B → (A × {x}) = (A × {B}))
3 feq1 4973 . . . 4 ((A × {x}) = (A × {B}) → ((A × {x}):A⟶{x} ↔ (A × {B}):A⟶{x}))
4 feq3 4975 . . . 4 ({x} = {B} → ((A × {B}):A⟶{x} ↔ (A × {B}):A⟶{B}))
53, 4sylan9bb 435 . . 3 (((A × {x}) = (A × {B}) {x} = {B}) → ((A × {x}):A⟶{x} ↔ (A × {B}):A⟶{B}))
62, 1, 5syl2anc 391 . 2 (x = B → ((A × {x}):A⟶{x} ↔ (A × {B}):A⟶{B}))
7 vex 2554 . . 3 x V
87fconst 5025 . 2 (A × {x}):A⟶{x}
96, 8vtoclg 2607 1 (B 𝑉 → (A × {B}):A⟶{B})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367   × cxp 4286  ⟶wf 4841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849 This theorem is referenced by:  fnconstg  5027  fconst6g  5028  xpsng  5281  fvconst2g  5318  fconst2g  5319
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