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Theorem fvdiagfn 8916
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 4642 . . 3 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32xpeq2d 5712 . 2 (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4 simpr 483 . 2 ((𝐼𝑊𝑋𝐵) → 𝑋𝐵)
5 snex 5437 . . . 4 {𝑋} ∈ V
6 xpexg 7758 . . . 4 ((𝐼𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
75, 6mpan2 689 . . 3 (𝐼𝑊 → (𝐼 × {𝑋}) ∈ V)
87adantr 479 . 2 ((𝐼𝑊𝑋𝐵) → (𝐼 × {𝑋}) ∈ V)
91, 3, 4, 8fvmptd3 7033 1 ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  {csn 4632  cmpt 5235   × cxp 5680  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561
This theorem is referenced by:  pwsdiagmhm  18790  pwsdiaglmhm  20949
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