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Theorem fvdiagfn 8252
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 fdiagfn.f . 2 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
2 sneq 4446 . . 3 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32xpeq2d 5434 . 2 (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4 simpr 477 . 2 ((𝐼𝑊𝑋𝐵) → 𝑋𝐵)
5 snex 5185 . . . 4 {𝑋} ∈ V
6 xpexg 7289 . . . 4 ((𝐼𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
75, 6mpan2 679 . . 3 (𝐼𝑊 → (𝐼 × {𝑋}) ∈ V)
87adantr 473 . 2 ((𝐼𝑊𝑋𝐵) → (𝐼 × {𝑋}) ∈ V)
91, 3, 4, 8fvmptd3 6616 1 ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3410  {csn 4436  cmpt 5005   × cxp 5402  cfv 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194
This theorem is referenced by:  pwsdiagmhm  17850  pwsdiaglmhm  19564
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