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Theorem ecexr 7956
Description: A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)

Proof of Theorem ecexr
StepHypRef Expression
1 n0i 4086 . . 3 (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2 snprc 4410 . . . . 5 𝐵 ∈ V ↔ {𝐵} = ∅)
3 imaeq2 5646 . . . . 5 ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
42, 3sylbi 208 . . . 4 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5 ima0 5665 . . . 4 (𝑅 “ ∅) = ∅
64, 5syl6eq 2815 . . 3 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
71, 6nsyl2 144 . 2 (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8 df-ec 7953 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
97, 8eleq2s 2862 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  c0 4081  {csn 4336  cima 5282  [cec 7949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-xp 5285  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ec 7953
This theorem is referenced by:  relelec  7994  ecdmn0  7996  erdisj  8001  eqvreldisj  34806
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