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Theorem bj-iminvval2 34760
 Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)
Hypotheses
Ref Expression
bj-iminvval2.exa (𝜑𝐴𝑈)
bj-iminvval2.exb (𝜑𝐵𝑉)
bj-iminvval2.arg (𝜑𝑅 ⊆ (𝐴 × 𝐵))
Assertion
Ref Expression
bj-iminvval2 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
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-iminvval2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 bj-iminvval2.exa . . 3 (𝜑𝐴𝑈)
2 bj-iminvval2.exb . . 3 (𝜑𝐵𝑉)
31, 2bj-iminvval 34759 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))}))
4 simpr 488 . . . . . . 7 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54cnveqd 5714 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
65imaeq1d 5899 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑦) = (𝑅𝑦))
76eqeq2d 2809 . . . 4 ((𝜑𝑟 = 𝑅) → (𝑥 = (𝑟𝑦) ↔ 𝑥 = (𝑅𝑦)))
87anbi2d 631 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦)) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))))
98opabbidv 5100 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
101, 2xpexd 7467 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
11 bj-iminvval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
1210, 11sselpwd 5198 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
131, 2bj-imdirval2lem 34748 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))} ∈ V)
143, 9, 12, 13fvmptd 6762 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ⊆ wss 3883  𝒫 cpw 4500  {copab 5096   × cxp 5521  ◡ccnv 5522   “ cima 5526  ‘cfv 6332  (class class class)co 7145  𝒫*ciminv 34757 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-iminv 34758 This theorem is referenced by:  bj-iminvid  34761
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