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Theorem dfse2 5934
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 5483 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
2 dfrab3 4233 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
3 iniseg 5931 . . . . . . 7 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥})
43elv 3449 . . . . . 6 (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥}
54ineq2i 4139 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
62, 5eqtr4i 2827 . . . 4 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ (𝑅 “ {𝑥}))
76eleq1i 2883 . . 3 ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
87ralbii 3136 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
91, 8bitri 278 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  {cab 2779  wral 3109  {crab 3113  Vcvv 3444  cin 3883  {csn 4528   class class class wbr 5033   Se wse 5480  ccnv 5522  cima 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-se 5483  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536
This theorem is referenced by:  isoselem  7077  fnse  7814
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