Theorem dmmpt 6243

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 5887	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		dfrn4 6203	. 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
3		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	mptpreima 6241	. 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	1, 2, 4	3eqtri 2792	1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  dmmptss  6244  dmmptg  6245  dmmptd  6684  fvmpti  6992  funcnvmpt  6995  fvmptss  7006  fvmptss2  7020  mptexgf  7227  tz9.12lem3  9768  cardf2  9945  pmtrsn  19633  00lsp  21152  rgrx0ndm  30001  abrexexd  32926  mptctf  33131  issibf  34788  rdgprc0  36320  imageval  36457  dmmptdff  45997  dmmptssf  46005  dmmptdf2  46006  dvcosre  46684  itgsinexplem1  46726  stirlinglem14  46859  fvmptrabdm  48088