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Theorem mptiniseg 6086
Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptiniseg (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
Distinct variable groups:   𝑥,𝐶   𝑥,𝑉
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mptiniseg
StepHypRef Expression
1 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21mptpreima 6085 . 2 (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 ∈ {𝐶}}
3 elsn2g 4593 . . 3 (𝐶𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
43rabbidv 3478 . 2 (𝐶𝑉 → {𝑥𝐴𝐵 ∈ {𝐶}} = {𝑥𝐴𝐵 = 𝐶})
52, 4syl5eq 2865 1 (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  {csn 4557  cmpt 5137  ccnv 5547  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  ramub1lem1  16350  frlmsslss  20846  symgtgp  22637  csscld  23779  clsocv  23780  sqff1o  25686  dchrfi  25758  poimirlem30  34803  ftc1anclem6  34853  pwssplit4  39567  pwslnmlem2  39571
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