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

Theorem fvdiagfn 6555
Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
fdiagfn.f 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
Assertion
Ref Expression
fvdiagfn ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐼   𝑥,𝑊   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem fvdiagfn
StepHypRef Expression
1 simpr 109 . 2 ((𝐼𝑊𝑋𝐵) → 𝑋𝐵)
2 snexg 4078 . . 3 (𝑋𝐵 → {𝑋} ∈ V)
3 xpexg 4623 . . 3 ((𝐼𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
42, 3sylan2 284 . 2 ((𝐼𝑊𝑋𝐵) → (𝐼 × {𝑋}) ∈ V)
5 sneq 3508 . . . 4 (𝑥 = 𝑋 → {𝑥} = {𝑋})
65xpeq2d 4533 . . 3 (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
7 fdiagfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
86, 7fvmptg 5465 . 2 ((𝑋𝐵 ∧ (𝐼 × {𝑋}) ∈ V) → (𝐹𝑋) = (𝐼 × {𝑋}))
91, 4, 8syl2anc 408 1 ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  {csn 3497  cmpt 3959   × cxp 4507  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator