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Theorem bj-imdirval2 34559
Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
Hypotheses
Ref Expression
bj-imdirval2.exa (𝜑𝐴𝑈)
bj-imdirval2.exb (𝜑𝐵𝑉)
bj-imdirval2.arg (𝜑𝑅 ⊆ (𝐴 × 𝐵))
Assertion
Ref Expression
bj-imdirval2 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-imdirval2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 bj-imdirval2.exa . . 3 (𝜑𝐴𝑈)
2 bj-imdirval2.exb . . 3 (𝜑𝐵𝑉)
31, 2bj-imdirval 34557 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
4 simpr 488 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54imaeq1d 5906 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑥) = (𝑅𝑥))
65eqeq1d 2824 . . . 4 ((𝜑𝑟 = 𝑅) → ((𝑟𝑥) = 𝑦 ↔ (𝑅𝑥) = 𝑦))
76anbi2d 631 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)))
87opabbidv 5108 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
91, 2xpexd 7459 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
10 bj-imdirval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
119, 10sselpwd 5206 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
121, 2bj-imdirval2lem 34558 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)} ∈ V)
133, 8, 11, 12fvmptd 6757 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  𝒫 cpw 4511  {copab 5104   × cxp 5530  cima 5535  cfv 6334  (class class class)co 7140  𝒫*cimdir 34554
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-imdir 34555
This theorem is referenced by:  bj-imdirval3  34560  bj-imdirid  34562  bj-imdirco  34566
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