Theorem dmmpt 6097

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 5767	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		dfrn4 6062	. 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
3		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	mptpreima 6095	. 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	1, 2, 4	3eqtri 2851	1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Syntax hints:   = wceq 1536   ∈ wcel 2113  {crab 3145  Vcvv 3497   ↦ cmpt 5149  ^◡ccnv 5557  dom cdm 5558  ran crn 5559   “ cima 5561

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-mpt 5150  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571

This theorem is referenced by:  dmmptss  6098  dmmptg  6099  dmmptd  6496  fvmpti  6770  fvmptss  6783  fvmptss2  6796  mptexgf  6988  tz9.12lem3  9221  cardf2  9375  pmtrsn  18650  00lsp  19756  rgrx0ndm  27378  abrexexd  30272  funcnvmpt  30415  mptctf  30456  issibf  31595  rdgprc0  33042  imageval  33395  dmmptdf  41494  dmmptssf  41508  dmmptdf2  41509  dvcosre  42202  itgsinexplem1  42245  stirlinglem14  42379  fvmptrabdm  43499