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Theorem fdiagfn 8903
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 6781 . . . 4 (𝑥𝐵 → (𝐼 × {𝑥}):𝐼𝐵)
21adantl 481 . . 3 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → (𝐼 × {𝑥}):𝐼𝐵)
3 elmapg 8852 . . . 4 ((𝐵𝑉𝐼𝑊) → ((𝐼 × {𝑥}) ∈ (𝐵m 𝐼) ↔ (𝐼 × {𝑥}):𝐼𝐵))
43adantr 480 . . 3 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → ((𝐼 × {𝑥}) ∈ (𝐵m 𝐼) ↔ (𝐼 × {𝑥}):𝐼𝐵))
52, 4mpbird 257 . 2 (((𝐵𝑉𝐼𝑊) ∧ 𝑥𝐵) → (𝐼 × {𝑥}) ∈ (𝐵m 𝐼))
6 fdiagfn.f . 2 𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))
75, 6fmptd 7119 1 ((𝐵𝑉𝐼𝑊) → 𝐹:𝐵⟶(𝐵m 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {csn 4625  cmpt 5226   × cxp 5671  wf 6539  (class class class)co 7415  m cmap 8839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8841
This theorem is referenced by:  pwsdiagmhm  18777
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