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Theorem seex 5213
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 5210 . 2 (𝑅 Se 𝐴 ↔ ∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V)
2 breq2 4791 . . . . 5 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
32rabbidv 3339 . . . 4 (𝑦 = 𝐵 → {𝑥𝐴𝑥𝑅𝑦} = {𝑥𝐴𝑥𝑅𝐵})
43eleq1d 2835 . . 3 (𝑦 = 𝐵 → ({𝑥𝐴𝑥𝑅𝑦} ∈ V ↔ {𝑥𝐴𝑥𝑅𝐵} ∈ V))
54rspccva 3459 . 2 ((∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V ∧ 𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
61, 5sylanb 570 1 ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351   class class class wbr 4787   Se wse 5207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-se 5210
This theorem is referenced by:  wereu2  5247  setlikespec  5843  fnse  7449  ordtypelem10  8592  frpomin  32075
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