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Theorem seex 5643
Description: The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
Assertion
Ref Expression
seex ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem seex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-se 5637 . 2 (𝑅 Se 𝐴 ↔ ∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V)
2 breq2 5146 . . . . 5 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
32rabbidv 3443 . . . 4 (𝑦 = 𝐵 → {𝑥𝐴𝑥𝑅𝑦} = {𝑥𝐴𝑥𝑅𝐵})
43eleq1d 2825 . . 3 (𝑦 = 𝐵 → ({𝑥𝐴𝑥𝑅𝑦} ∈ V ↔ {𝑥𝐴𝑥𝑅𝐵} ∈ V))
54rspccva 3620 . 2 ((∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V ∧ 𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
61, 5sylanb 581 1 ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479   class class class wbr 5142   Se wse 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-se 5637
This theorem is referenced by:  wereu2  5681  setlikespec  6345  frpomin  6360  fnse  8159  ordtypelem10  9568  weiunse  36470
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