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Theorem intimasn 43670
Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimasn (𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎 “ {𝐵})})
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝑉(𝑥,𝑎)

Proof of Theorem intimasn
Dummy variables 𝑦 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-5 1910 . 2 (𝐵𝑉 → ∀𝑦 𝐵𝑉)
2 r19.12sn 4720 . . . 4 (𝐵𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎))
32biimprd 248 . . 3 (𝐵𝑉 → (∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
43alimi 1811 . 2 (∀𝑦 𝐵𝑉 → ∀𝑦(∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
5 intimag 43669 . 2 (∀𝑦(∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴 “ {𝐵}) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎 “ {𝐵})})
61, 4, 53syl 18 1 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎 “ {𝐵})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {csn 4626  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-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  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:  intimasn2  43671  brtrclfv2  43740
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