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Theorem fvdiagfn 8884
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 4633 . . 3 (𝑥 = 𝑋 → {𝑥} = {𝑋})
32xpeq2d 5699 . 2 (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋}))
4 simpr 484 . 2 ((𝐼𝑊𝑋𝐵) → 𝑋𝐵)
5 snex 5424 . . . 4 {𝑋} ∈ V
6 xpexg 7733 . . . 4 ((𝐼𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V)
75, 6mpan2 688 . . 3 (𝐼𝑊 → (𝐼 × {𝑋}) ∈ V)
87adantr 480 . 2 ((𝐼𝑊𝑋𝐵) → (𝐼 × {𝑋}) ∈ V)
91, 3, 4, 8fvmptd3 7014 1 ((𝐼𝑊𝑋𝐵) → (𝐹𝑋) = (𝐼 × {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623  cmpt 5224   × cxp 5667  cfv 6536
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544
This theorem is referenced by:  pwsdiagmhm  18753  pwsdiaglmhm  20902
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