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Theorem mptiniseg 5773
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 5772 . 2 (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 ∈ {𝐶}}
3 elsn2g 4349 . . 3 (𝐶𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
43rabbidv 3339 . 2 (𝐶𝑉 → {𝑥𝐴𝐵 ∈ {𝐶}} = {𝑥𝐴𝐵 = 𝐶})
52, 4syl5eq 2817 1 (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  {csn 4316  cmpt 4863  ccnv 5248  cima 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  ramub1lem1  15937  frlmsslss  20330  symgtgp  22125  csscld  23267  clsocv  23268  sqff1o  25129  dchrfi  25201  poimirlem30  33772  ftc1anclem6  33822  pwssplit4  38185  pwslnmlem2  38189
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