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Theorem intimasn 39987
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 1904 . 2 (𝐵𝑉 → ∀𝑦 𝐵𝑉)
2 r19.12sn 4648 . . . 4 (𝐵𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎))
32biimprd 250 . . 3 (𝐵𝑉 → (∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
43alimi 1805 . 2 (∀𝑦 𝐵𝑉 → ∀𝑦(∀𝑎𝐴𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
5 intimag 39986 . 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 1528   = wceq 1530  wcel 2107  {cab 2797  wral 3136  wrex 3137  {csn 4559  cop 4565   cint 4867  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  intimasn2  39988  brtrclfv2  40057
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