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Theorem dfse2 4698
 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 4070 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
2 dfrab3 3213 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
3 vex 2560 . . . . . . 7 𝑥 ∈ V
4 iniseg 4697 . . . . . . 7 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥})
53, 4ax-mp 7 . . . . . 6 (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥}
65ineq2i 3135 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
72, 6eqtr4i 2063 . . . 4 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ (𝑅 “ {𝑥}))
87eleq1i 2103 . . 3 ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
98ralbii 2330 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
101, 9bitri 173 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  {crab 2310  Vcvv 2557   ∩ cin 2916  {csn 3375   class class class wbr 3764   Se wse 4066  ◡ccnv 4344   “ cima 4348 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-se 4070  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  isoselem  5459
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