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Theorem elimasni 5955
Description: Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
Assertion
Ref Expression
elimasni (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)

Proof of Theorem elimasni
StepHypRef Expression
1 noel 4295 . . . . 5 ¬ 𝐶 ∈ ∅
2 snprc 4652 . . . . . . . . 9 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 218 . . . . . . . 8 𝐵 ∈ V → {𝐵} = ∅)
43imaeq2d 5928 . . . . . . 7 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅))
5 ima0 5944 . . . . . . 7 (𝐴 “ ∅) = ∅
64, 5syl6eq 2872 . . . . . 6 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅)
76eleq2d 2898 . . . . 5 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅))
81, 7mtbiri 329 . . . 4 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵}))
98con4i 114 . . 3 (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V)
10 elex 3512 . . 3 (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V)
119, 10jca 514 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
12 elimasng 5954 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
13 df-br 5066 . . . 4 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
1412, 13syl6bbr 291 . . 3 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
1514biimpd 231 . 2 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶))
1611, 15mpcom 38 1 (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  c0 4290  {csn 4566  cop 4572   class class class wbr 5065  cima 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  dffv2  6755  poimirlem2  34893  poimirlem23  34914
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