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Theorem intima0 37758
 Description: Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intima0 𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑎
Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑎)

Proof of Theorem intima0
StepHypRef Expression
1 vex 3198 . . 3 𝑎 ∈ V
2 imaexg 7088 . . 3 (𝑎 ∈ V → (𝑎𝐵) ∈ V)
31, 2ax-mp 5 . 2 (𝑎𝐵) ∈ V
43dfiin2 4546 1 𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  {cab 2606  ∃wrex 2910  Vcvv 3195  ∩ cint 4466  ∩ ciin 4512   “ cima 5107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iin 4514  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117 This theorem is referenced by:  intimass2  37766  intimasn2  37769
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