Theorem mptiniseg 6259

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 6258	. 2 ⊢ (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}}
3		elsn2g 4664	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
4	3	rabbidv 3444	. 2 ⊢ (𝐶 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}} = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})
5	2, 4	eqtrid 2789	1 ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {crab 3436  {csn 4626   ↦ cmpt 5225  ^◡ccnv 5684   “ cima 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  ramub1lem1  17064  frlmsslss  21794  symgtgp  24114  csscld  25283  clsocv  25284  sqff1o  27225  dchrfi  27299  poimirlem30  37657  ftc1anclem6  37705  pwssplit4  43101  pwslnmlem2  43105