Theorem mptiniseg 6142

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 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
mptiniseg	⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})

Distinct variable groups:   𝑥,𝐶   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mptiniseg

Step	Hyp	Ref	Expression
1		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	mptpreima 6141	. 2 ⊢ (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}}
3		elsn2g 4599	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
4	3	rabbidv 3414	. 2 ⊢ (𝐶 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}} = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})
5	2, 4	eqtrid 2790	1 ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  {crab 3068  {csn 4561   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  ramub1lem1  16727  frlmsslss  20981  symgtgp  23257  csscld  24413  clsocv  24414  sqff1o  26331  dchrfi  26403  poimirlem30  35807  ftc1anclem6  35855  pwssplit4  40914  pwslnmlem2  40918