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Theorem intimag 43669
Description: Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimag (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
Distinct variable groups:   𝑥,𝑎,𝑦,𝐴   𝐵,𝑎,𝑥,𝑦   𝐴,𝑏   𝐵,𝑏,𝑎,𝑦,𝑥

Proof of Theorem intimag
StepHypRef Expression
1 r19.12 3314 . . . . 5 (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
2 id 22 . . . . 5 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
31, 2impbid2 226 . . . 4 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
4 elimaint 43662 . . . 4 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
5 elintima 43666 . . . 4 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
63, 4, 53bitr4g 314 . . 3 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
76alimi 1811 . 2 (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ∀𝑦(𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
8 dfcleq 2730 . 2 (( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑦(𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
97, 8sylibr 234 1 (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  cop 4632   cint 4946  cima 5688
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-sep 5296  ax-nul 5306  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-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  intimasn  43670
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