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Theorem dfse3 6359
Description: Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.)
Assertion
Ref Expression
dfse3 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfse3
StepHypRef Expression
1 dfse2 6121 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
2 df-pred 6323 . . . 4 Pred(𝑅, 𝐴, 𝑥) = (𝐴 ∩ (𝑅 “ {𝑥}))
32eleq1i 2830 . . 3 (Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
43ralbii 3091 . 2 (∀𝑥𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
51, 4bitr4i 278 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  wral 3059  Vcvv 3478  cin 3962  {csn 4631   Se wse 5639  ccnv 5688  cima 5692  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-se 5642  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  sexp2  8170  sexp3  8177  ttrclselem2  9764  ttrclse  9765
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