Theorem mptiniseg 6196

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 6195	. 2 ⊢ (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}}
3		elsn2g 4629	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
4	3	rabbidv 3413	. 2 ⊢ (𝐶 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}} = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})
5	2, 4	eqtrid 2783	1 ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  {crab 3405  {csn 4591   ↦ cmpt 5193  ^◡ccnv 5637   “ cima 5641

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 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-mpt 5194  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  ramub1lem1  16909  frlmsslss  21217  symgtgp  23494  csscld  24650  clsocv  24651  sqff1o  26568  dchrfi  26640  poimirlem30  36181  ftc1anclem6  36229  pwssplit4  41474  pwslnmlem2  41478