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Theorem bj-iminvval2 36013
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 36012 . 2 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))}))
4 simpr 486 . . . . . . 7 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
54cnveqd 5873 . . . . . 6 ((𝜑𝑟 = 𝑅) → 𝑟 = 𝑅)
65imaeq1d 6056 . . . . 5 ((𝜑𝑟 = 𝑅) → (𝑟𝑦) = (𝑅𝑦))
76eqeq2d 2744 . . . 4 ((𝜑𝑟 = 𝑅) → (𝑥 = (𝑟𝑦) ↔ 𝑥 = (𝑅𝑦)))
87anbi2d 630 . . 3 ((𝜑𝑟 = 𝑅) → (((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦)) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))))
98opabbidv 5213 . 2 ((𝜑𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑟𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
101, 2xpexd 7733 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
11 bj-iminvval2.arg . . 3 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
1210, 11sselpwd 5325 . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
131, 2bj-imdirval2lem 36001 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))} ∈ V)
143, 9, 12, 13fvmptd 7001 1 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑥 = (𝑅𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  wss 3947  𝒫 cpw 4601  {copab 5209   × cxp 5673  ccnv 5674  cima 5678  cfv 6540  (class class class)co 7404  𝒫*ciminv 36010
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-iminv 36011
This theorem is referenced by:  bj-iminvid  36014
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