Theorem dmmpt 6260

Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Hypothesis

Ref	Expression
dmmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
dmmpt	⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Proof of Theorem dmmpt

Step	Hyp	Ref	Expression
1		dfdm4 5906	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		dfrn4 6222	. 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
3		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	mptpreima 6258	. 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	1, 2, 4	3eqtri 2769	1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Syntax hints:   = wceq 1540   ∈ wcel 2108  {crab 3436  Vcvv 3480   ↦ cmpt 5225  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ 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:  dmmptss  6261  dmmptg  6262  dmmptd  6713  fvmpti  7015  fvmptss  7028  fvmptss2  7042  mptexgf  7242  tz9.12lem3  9829  cardf2  9983  pmtrsn  19537  00lsp  20979  rgrx0ndm  29611  abrexexd  32528  funcnvmpt  32677  mptctf  32729  issibf  34335  rdgprc0  35794  imageval  35931  dmmptdff  45228  dmmptssf  45237  dmmptdf2  45238  dvcosre  45927  itgsinexplem1  45969  stirlinglem14  46102  fvmptrabdm  47305