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Theorem intimasn 38556
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 2005 . 2 (𝐵𝑉 → ∀𝑦 𝐵𝑉)
2 r19.12sn 4410 . . . 4 (𝐵𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎))
32biimprd 239 . . 3 (𝐵𝑉 → (∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
43alimi 1906 . 2 (∀𝑦 𝐵𝑉 → ∀𝑦(∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
5 intimag 38555 . 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 1650   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  {csn 4334  cop 4340   cint 4633  cima 5280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290
This theorem is referenced by:  intimasn2  38557  brtrclfv2  38626
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