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Theorem intimasn2 36969
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 36968 . 2 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})})
2 intima0 36958 . 2 𝑥𝐴 (𝑥 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})}
31, 2syl6eqr 2662 1 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  {csn 4125   cint 4410   ciin 4456  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by: (None)
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