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Theorem bj-iminvval2 36565
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 36564 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))}))
4 simpr 484 . . . . . . 7 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54cnveqd 5865 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
65imaeq1d 6048 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑦) = (𝑅𝑦))
76eqeq2d 2735 . . . 4 ((𝜑𝑟 = 𝑅) → (𝑥 = (𝑟𝑦) ↔ 𝑥 = (𝑅𝑦)))
87anbi2d 628 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦)) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))))
98opabbidv 5204 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
101, 2xpexd 7731 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
11 bj-iminvval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
1210, 11sselpwd 5316 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
131, 2bj-imdirval2lem 36553 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))} ∈ V)
143, 9, 12, 13fvmptd 6995 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940  𝒫 cpw 4594  {copab 5200   × cxp 5664  ccnv 5665  cima 5669  cfv 6533  (class class class)co 7401  𝒫*ciminv 36562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-iminv 36563
This theorem is referenced by:  bj-iminvid  36566
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