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Theorem intima0 39983
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 3496 . . 3 𝑎 ∈ V
21imaex 7613 . 2 (𝑎𝐵) ∈ V
32dfiin2 4950 1 𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1531  {cab 2797  wrex 3137   cint 4867   ciin 4911  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  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 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  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-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-iin 4913  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:  intimass2  39991  intimasn2  39994
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