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Theorem elrelimasn 4995
Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
elrelimasn (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elrelimasn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elimag 4975 . . . . . 6 (𝐵 ∈ (𝑅 “ {𝐴}) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵))
21ibi 176 . . . . 5 (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵)
3 rexm 3523 . . . . 5 (∃𝑦 ∈ {𝐴}𝑦𝑅𝐵 → ∃𝑦 𝑦 ∈ {𝐴})
4 elsni 3611 . . . . . 6 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
54eximi 1600 . . . . 5 (∃𝑦 𝑦 ∈ {𝐴} → ∃𝑦 𝑦 = 𝐴)
62, 3, 53syl 17 . . . 4 (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 𝑦 = 𝐴)
7 isset 2744 . . . 4 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
86, 7sylibr 134 . . 3 (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V)
98a1i 9 . 2 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V))
10 brrelex1 4666 . . 3 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
1110ex 115 . 2 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
12 imasng 4994 . . . . 5 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
1312eleq2d 2247 . . . 4 (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
14 brrelex2 4668 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
1514ex 115 . . . . 5 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
16 breq2 4008 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
1716elab3g 2889 . . . . 5 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
1815, 17syl 14 . . . 4 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
1913, 18sylan9bbr 463 . . 3 ((Rel 𝑅𝐴 ∈ V) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
2019ex 115 . 2 (Rel 𝑅 → (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)))
219, 11, 20pm5.21ndd 705 1 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wrex 2456  Vcvv 2738  {csn 3593   class class class wbr 4004  cima 4630  Rel wrel 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640
This theorem is referenced by:  eliniseg2  5009
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