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Theorem bj-imdirval 34490
Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
Hypotheses
Ref Expression
bj-imdirval.1 (𝜑𝐴𝑈)
bj-imdirval.2 (𝜑𝐵𝑉)
Assertion
Ref Expression
bj-imdirval (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
Distinct variable groups:   𝐴,𝑟,𝑥,𝑦   𝐵,𝑟,𝑥,𝑦   𝜑,𝑟
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦,𝑟)   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem bj-imdirval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-imdir 34489 . . 3 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)}))
21a1i 11 . 2 (𝜑 → 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)})))
3 xpeq12 5553 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
43pweqd 4531 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
54adantl 485 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6 sseq2 3969 . . . . . . 7 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
7 sseq2 3969 . . . . . . 7 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
86, 7bi2anan9 638 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑥𝑎𝑦𝑏) ↔ (𝑥𝐴𝑦𝐵)))
98anbi1d 632 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)))
109opabbidv 5105 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)})
1110adantl 485 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)})
125, 11mpteq12dv 5124 . 2 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)}) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
13 bj-imdirval.1 . . 3 (𝜑𝐴𝑈)
1413elexd 3491 . 2 (𝜑𝐴 ∈ V)
15 bj-imdirval.2 . . 3 (𝜑𝐵𝑉)
1615elexd 3491 . 2 (𝜑𝐵 ∈ V)
1713, 15xpexd 7449 . . . 4 (𝜑 → (𝐴 × 𝐵) ∈ V)
1817pwexd 5253 . . 3 (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
1918mptexd 6960 . 2 (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}) ∈ V)
202, 12, 14, 16, 19ovmpod 7276 1 (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  wss 3910  𝒫 cpw 4512  {copab 5101  cmpt 5119   × cxp 5526  cima 5531  (class class class)co 7130  cmpo 7132  𝒫*cimdir 34488
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-imdir 34489
This theorem is referenced by:  bj-imdirval2  34491
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