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Theorem mptiniseg 6072
 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 6071 . 2 (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 ∈ {𝐶}}
3 elsn2g 4563 . . 3 (𝐶𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
43rabbidv 3392 . 2 (𝐶𝑉 → {𝑥𝐴𝐵 ∈ {𝐶}} = {𝑥𝐴𝐵 = 𝐶})
52, 4syl5eq 2805 1 (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3074  {csn 4525   ↦ cmpt 5115  ◡ccnv 5526   “ cima 5530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-mpt 5116  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540 This theorem is referenced by:  ramub1lem1  16422  frlmsslss  20544  symgtgp  22811  csscld  23954  clsocv  23955  sqff1o  25871  dchrfi  25943  poimirlem30  35393  ftc1anclem6  35441  pwssplit4  40434  pwslnmlem2  40438
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