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Theorem dfse2 4848
Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfse2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfse2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-se 4193 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
2 dfrab3 3299 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
3 vex 2644 . . . . . . 7 𝑥 ∈ V
4 iniseg 4847 . . . . . . 7 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥})
53, 4ax-mp 7 . . . . . 6 (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥}
65ineq2i 3221 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
72, 6eqtr4i 2123 . . . 4 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ (𝑅 “ {𝑥}))
87eleq1i 2165 . . 3 ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
98ralbii 2400 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
101, 9bitri 183 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1299  wcel 1448  {cab 2086  wral 2375  {crab 2379  Vcvv 2641  cin 3020  {csn 3474   class class class wbr 3875   Se wse 4189  ccnv 4476  cima 4480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-se 4193  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490
This theorem is referenced by:  isoselem  5653
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