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Theorem seex 5512
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 5509 . 2 (𝑅 Se 𝐴 ↔ ∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V)
2 breq2 5062 . . . . 5 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
32rabbidv 3481 . . . 4 (𝑦 = 𝐵 → {𝑥𝐴𝑥𝑅𝑦} = {𝑥𝐴𝑥𝑅𝐵})
43eleq1d 2897 . . 3 (𝑦 = 𝐵 → ({𝑥𝐴𝑥𝑅𝑦} ∈ V ↔ {𝑥𝐴𝑥𝑅𝐵} ∈ V))
54rspccva 3621 . 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 396   = wceq 1528  wcel 2105  wral 3138  {crab 3142  Vcvv 3495   class class class wbr 5058   Se wse 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-se 5509
This theorem is referenced by:  wereu2  5546  setlikespec  6163  fnse  7818  ordtypelem10  8980  frpomin  32976
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