Theorem dmmpt 5875

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 5552	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		dfrn4 5840	. 2 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
3		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	mptpreima 5873	. 2 ⊢ (◡𝐹 “ V) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	1, 2, 4	3eqtri 2853	1 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Syntax hints:   = wceq 1656   ∈ wcel 2164  {crab 3121  Vcvv 3414   ↦ cmpt 4954  ^◡ccnv 5345  dom cdm 5346  ran crn 5347   “ cima 5349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-mpt 4955  df-xp 5352  df-rel 5353  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359

This theorem is referenced by:  dmmptss  5876  dmmptg  5877  dmmptd  6261  fvmpti  6532  fvmptss  6544  fvmptss2  6557  mptexgf  6746  tz9.12lem3  8936  cardf2  9089  pmtrsn  18297  00lsp  19347  rgrx0ndm  26898  abrexexd  29891  funcnvmpt  30012  mptctf  30039  issibf  30936  rdgprc0  32232  imageval  32571  dmmptdf  40218  dmmptssf  40235  dmmptdf2  40236  dvcosre  40915  itgsinexplem1  40958  stirlinglem14  41092  fvmptrabdm  42190