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Theorem dfima3 5925
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfima3 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem dfima3
StepHypRef Expression
1 dfima2 5924 . 2 (𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
2 df-br 5058 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
32rexbii 3245 . . . 4 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴)
4 df-rex 3142 . . . 4 (∃𝑥𝐵𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
53, 4bitri 277 . . 3 (∃𝑥𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
65abbii 2884 . 2 {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
71, 6eqtri 2842 1 (𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1530  wex 1773  wcel 2107  {cab 2797  wrex 3137  cop 4565   class class class wbr 5057  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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-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:  imadmrn  5932  imassrn  5933  imai  5935  funimaexg  6433  cnvimadfsn  7831  rdglim2  8060  dfhe3  40106
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