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Theorem mptiniseg 6238
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 6237 . 2 (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 ∈ {𝐶}}
3 elsn2g 4666 . . 3 (𝐶𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
43rabbidv 3440 . 2 (𝐶𝑉 → {𝑥𝐴𝐵 ∈ {𝐶}} = {𝑥𝐴𝐵 = 𝐶})
52, 4eqtrid 2784 1 (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3432  {csn 4628  cmpt 5231  ccnv 5675  cima 5679
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  ramub1lem1  16963  frlmsslss  21548  symgtgp  23830  csscld  24990  clsocv  24991  sqff1o  26910  dchrfi  26982  poimirlem30  36821  ftc1anclem6  36869  pwssplit4  42133  pwslnmlem2  42137
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