MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvdiagfn Structured version   Visualization version   GIF version

Theorem fvdiagfn 7846
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 477 . 2 ((𝐼𝑊𝑋𝐵) → 𝑋𝐵)
2 snex 4869 . . . 4 {𝑋} ∈ V
3 xpexg 6913 . . . 4 ((𝐼𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
42, 3mpan2 706 . . 3 (𝐼𝑊 → (𝐼 × {𝑋}) ∈ V)
54adantr 481 . 2 ((𝐼𝑊𝑋𝐵) → (𝐼 × {𝑋}) ∈ V)
6 sneq 4158 . . . 4 (𝑥 = 𝑋 → {𝑥} = {𝑋})
76xpeq2d 5099 . . 3 (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
8 fdiagfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
97, 8fvmptg 6237 . 2 ((𝑋𝐵 ∧ (𝐼 × {𝑋}) ∈ V) → (𝐹𝑋) = (𝐼 × {𝑋}))
101, 5, 9syl2anc 692 1 ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  cmpt 4673   × cxp 5072  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  pwsdiagmhm  17290  pwsdiaglmhm  18976
  Copyright terms: Public domain W3C validator