ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfse2 Structured version   GIF version

Theorem dfse2 4640
Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfse2 (𝑅 Se Ax A (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 4055 . 2 (𝑅 Se Ax A {y Ay𝑅x} V)
2 dfrab3 3207 . . . . 5 {y Ay𝑅x} = (A ∩ {yy𝑅x})
3 vex 2554 . . . . . . 7 x V
4 iniseg 4639 . . . . . . 7 (x V → (𝑅 “ {x}) = {yy𝑅x})
53, 4ax-mp 7 . . . . . 6 (𝑅 “ {x}) = {yy𝑅x}
65ineq2i 3129 . . . . 5 (A ∩ (𝑅 “ {x})) = (A ∩ {yy𝑅x})
72, 6eqtr4i 2060 . . . 4 {y Ay𝑅x} = (A ∩ (𝑅 “ {x}))
87eleq1i 2100 . . 3 ({y Ay𝑅x} V ↔ (A ∩ (𝑅 “ {x})) V)
98ralbii 2324 . 2 (x A {y Ay𝑅x} V ↔ x A (A ∩ (𝑅 “ {x})) V)
101, 9bitri 173 1 (𝑅 Se Ax A (A ∩ (𝑅 “ {x})) V)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300  {crab 2304  Vcvv 2551  cin 2910  {csn 3366   class class class wbr 3754   Se wse 4054  ccnv 4286  cima 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-se 4055  df-xp 4293  df-cnv 4295  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300
This theorem is referenced by:  isoselem  5400
  Copyright terms: Public domain W3C validator