Theorem mptiniseg 6197

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 6196	. 2 ⊢ (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {𝐶}}
3		elsn2g 4621	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ {𝐶} ↔ 𝐵 = 𝐶))
4	3	rabbidv 3406	. 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 2113  {crab 3399  {csn 4580   ↦ cmpt 5179  ^◡ccnv 5623   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-mpt 5180  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  ramub1lem1  16954  frlmsslss  21729  symgtgp  24050  csscld  25205  clsocv  25206  sqff1o  27148  dchrfi  27222  poimirlem30  37851  ftc1anclem6  37899  pwssplit4  43331  pwslnmlem2  43335