Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fconstg GIF version

Theorem fconstg 5083
 Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
fconstg (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})

Proof of Theorem fconstg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 3386 . . . 4 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21xpeq2d 4369 . . 3 (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3 feq1 5030 . . . 4 ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4 feq3 5032 . . . 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 2560 . . 3 𝑥 ∈ V
87fconst 5082 . 2 (𝐴 × {𝑥}):𝐴⟶{𝑥}
96, 8vtoclg 2613 1 (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {csn 3375   × cxp 4343  ⟶wf 4898 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906 This theorem is referenced by:  fnconstg  5084  fconst6g  5085  xpsng  5338  fvconst2g  5375  fconst2g  5376
 Copyright terms: Public domain W3C validator