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Theorem intimasn2 40191
 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 40190 . 2 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})})
2 intima0 40180 . 2 𝑥𝐴 (𝑥 “ {𝐵}) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥 “ {𝐵})}
31, 2syl6eqr 2877 1 (𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  {cab 2802  ∃wrex 3133  {csn 4548  ∩ cint 4857  ∩ ciin 4901   “ cima 5539 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-int 4858  df-iin 4903  df-br 5048  df-opab 5110  df-xp 5542  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549 This theorem is referenced by: (None)
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