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Theorem bj-imdirval2 35350
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 35348 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
4 simpr 485 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54imaeq1d 5967 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑥) = (𝑅𝑥))
65eqeq1d 2742 . . . 4 ((𝜑𝑟 = 𝑅) → ((𝑟𝑥) = 𝑦 ↔ (𝑅𝑥) = 𝑦))
76anbi2d 629 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)))
87opabbidv 5145 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
91, 2xpexd 7595 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
10 bj-imdirval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
119, 10sselpwd 5254 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
121, 2bj-imdirval2lem 35349 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)} ∈ V)
133, 8, 11, 12fvmptd 6879 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  Vcvv 3431  wss 3892  𝒫 cpw 4539  {copab 5141   × cxp 5588  cima 5593  cfv 6432  (class class class)co 7271  𝒫*cimdir 35345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-imdir 35346
This theorem is referenced by:  bj-imdirval3  35351  bj-imdirid  35353  bj-imdirco  35357
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