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Theorem bj-imdirvallem 37181
Description: Lemma for bj-imdirval 37182 and bj-iminvval 37194. (Contributed by BJ, 23-May-2024.)
Hypotheses
Ref Expression
bj-imdirvallem.1 (𝜑𝐴𝑈)
bj-imdirvallem.2 (𝜑𝐵𝑉)
bj-imdirvallem.df 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)}))
Assertion
Ref Expression
bj-imdirvallem (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥,𝑦   𝐵,𝑎,𝑏,𝑟,𝑥,𝑦   𝜑,𝑎,𝑏,𝑟   𝜓,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑟)   𝐶(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem bj-imdirvallem
StepHypRef Expression
1 bj-imdirvallem.df . . 3 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)}))
21a1i 11 . 2 (𝜑𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)})))
3 xpeq12 5710 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
43pweqd 4617 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
54adantl 481 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6 sseq2 4010 . . . . . . 7 (𝑎 = 𝐴 → (𝑥𝑎𝑥𝐴))
7 sseq2 4010 . . . . . . 7 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
86, 7bi2anan9 638 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑥𝑎𝑦𝑏) ↔ (𝑥𝐴𝑦𝐵)))
98anbi1d 631 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑥𝑎𝑦𝑏) ∧ 𝜓) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)))
109opabbidv 5209 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})
1110adantl 481 . . 3 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})
125, 11mpteq12dv 5233 . 2 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝜓)}) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}))
13 bj-imdirvallem.1 . . 3 (𝜑𝐴𝑈)
1413elexd 3504 . 2 (𝜑𝐴 ∈ V)
15 bj-imdirvallem.2 . . 3 (𝜑𝐵𝑉)
1615elexd 3504 . 2 (𝜑𝐵 ∈ V)
1713, 15xpexd 7771 . . . 4 (𝜑 → (𝐴 × 𝐵) ∈ V)
1817pwexd 5379 . . 3 (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
1918mptexd 7244 . 2 (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}) ∈ V)
202, 12, 14, 16, 19ovmpod 7585 1 (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  𝒫 cpw 4600  {copab 5205  cmpt 5225   × cxp 5683  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  bj-imdirval  37182  bj-iminvval  37194
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