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Theorem elreimasng 4691
 Description: Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
Assertion
Ref Expression
elreimasng ((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elreimasng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imasng 4690 . . 3 (𝐴𝑉 → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
21eleq2d 2107 . 2 (𝐴𝑉 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
3 brrelex2 4383 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
43ex 108 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
5 breq2 3768 . . . 4 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
65elab3g 2693 . . 3 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
74, 6syl 14 . 2 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
82, 7sylan9bbr 436 1 ((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  {cab 2026  Vcvv 2557  {csn 3375   class class class wbr 3764   “ cima 4348  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
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