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Theorem fdiagfn 8437
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
fdiagfn ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵m 𝐼))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐼   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem fdiagfn
StepHypRef Expression
1 fconst6g 6549 . . . 4 (𝑥𝐵 → (𝐼 × {𝑥}):𝐼𝐵)
21adantl 485 . . 3 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → (𝐼 × {𝑥}):𝐼𝐵)
3 elmapg 8402 . . . 4 ((𝐵𝑉𝐼𝑊) → ((𝐼 × {𝑥}) ∈ (𝐵m 𝐼) ↔ (𝐼 × {𝑥}):𝐼𝐵))
43adantr 484 . . 3 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → ((𝐼 × {𝑥}) ∈ (𝐵m 𝐼) ↔ (𝐼 × {𝑥}):𝐼𝐵))
52, 4mpbird 260 . 2 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → (𝐼 × {𝑥}) ∈ (𝐵m 𝐼))
6 fdiagfn.f . 2 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
75, 6fmptd 6859 1 ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵m 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {csn 4548  cmpt 5127   × cxp 5534  wf 6332  (class class class)co 7138  m cmap 8389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391
This theorem is referenced by:  pwsdiagmhm  17984
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