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Theorem mptiniseg 4912
Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1 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 dmmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21mptpreima 4911 . 2 (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 ∈ {𝐶}}
3 elsn2g 3472 . . 3 (𝐶𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
43rabbidv 2608 . 2 (𝐶𝑉 → {𝑥𝐴𝐵 ∈ {𝐶}} = {𝑥𝐴𝐵 = 𝐶})
52, 4syl5eq 2132 1 (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  {crab 2363  {csn 3441  cmpt 3891  ccnv 4427  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by: (None)
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