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Theorem intimasn2 41834
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
intimasn2 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem intimasn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 intimasn 41833 . 2 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})})
2 intima0 41824 . 2 𝑥𝐴 (𝑥 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})}
31, 2eqtr4di 2795 1 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {cab 2714  wrex 3071  {csn 4584   cint 4905   ciin 4953  cima 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iin 4955  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by: (None)
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