Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-imdirval Structured version   Visualization version   GIF version

Theorem bj-imdirval 36660
Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
Hypotheses
Ref Expression
bj-imdirval.1 (𝜑𝐴𝑈)
bj-imdirval.2 (𝜑𝐵𝑉)
Assertion
Ref Expression
bj-imdirval (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
Distinct variable groups:   𝐴,𝑟,𝑥,𝑦   𝐵,𝑟,𝑥,𝑦   𝜑,𝑟
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦,𝑟)   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem bj-imdirval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-imdirval.1 . 2 (𝜑𝐴𝑈)
2 bj-imdirval.2 . 2 (𝜑𝐵𝑉)
3 df-imdir 36658 . 2 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)}))
41, 2, 3bj-imdirvallem 36659 1 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wss 3947  𝒫 cpw 4603  {copab 5210  cmpt 5231   × cxp 5676  cima 5681  (class class class)co 7420  𝒫*cimdir 36657
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
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 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-imdir 36658
This theorem is referenced by:  bj-imdirval2  36662
  Copyright terms: Public domain W3C validator