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Theorem elimasni 5527
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 3952 . . . . 5 ¬ 𝐶 ∈ ∅
2 snprc 4285 . . . . . . . . 9 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 206 . . . . . . . 8 𝐵 ∈ V → {𝐵} = ∅)
43imaeq2d 5501 . . . . . . 7 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅))
5 ima0 5516 . . . . . . 7 (𝐴 “ ∅) = ∅
64, 5syl6eq 2701 . . . . . 6 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅)
76eleq2d 2716 . . . . 5 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅))
81, 7mtbiri 316 . . . 4 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵}))
98con4i 113 . . 3 (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V)
10 elex 3243 . . 3 (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V)
119, 10jca 553 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
12 elimasng 5526 . . . 4 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
13 df-br 4686 . . . 4 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
1412, 13syl6bbr 278 . . 3 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
1514biimpd 219 . 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 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210  cop 4216   class class class wbr 4685  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  dffv2  6310  poimirlem2  33541  poimirlem23  33562
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