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Theorem bj-imdirval2 35354
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 35352 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
4 simpr 485 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54imaeq1d 5968 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑥) = (𝑅𝑥))
65eqeq1d 2740 . . . 4 ((𝜑𝑟 = 𝑅) → ((𝑟𝑥) = 𝑦 ↔ (𝑅𝑥) = 𝑦))
76anbi2d 629 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)))
87opabbidv 5140 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
91, 2xpexd 7601 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
10 bj-imdirval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
119, 10sselpwd 5250 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
121, 2bj-imdirval2lem 35353 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)} ∈ V)
133, 8, 11, 12fvmptd 6882 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  wss 3887  𝒫 cpw 4533  {copab 5136   × cxp 5587  cima 5592  cfv 6433  (class class class)co 7275  𝒫*cimdir 35349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-imdir 35350
This theorem is referenced by:  bj-imdirval3  35355  bj-imdirid  35357  bj-imdirco  35361
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