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Theorem dmmptd 6492
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
dmmptd.a 𝐴 = (𝑥𝐵𝐶)
dmmptd.c ((𝜑𝑥𝐵) → 𝐶𝑉)
Assertion
Ref Expression
dmmptd (𝜑 → dom 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem dmmptd
StepHypRef Expression
1 dmmptd.c . . . . 5 ((𝜑𝑥𝐵) → 𝐶𝑉)
21elexd 3520 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
32ralrimiva 3187 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
4 rabid2 3387 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
53, 4sylibr 235 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
6 dmmptd.a . . 3 𝐴 = (𝑥𝐵𝐶)
76dmmpt 6093 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
85, 7syl6reqr 2880 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cmpt 5143  dom cdm 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  lo1eq  14920  rlimeq  14921  rlimcld2  14930  rlimcn2  14942  rlimmptrcl  14959  rlimsqzlem  15000  dprdz  19088  alexsublem  22587  cmetcaulem  23825  minveclem3b  23965  mbfneg  24185  mbfsup  24199  mbfinf  24200  mbflimsup  24201  itg2monolem1  24285  itg2mono  24288  itg2i1fseq2  24291  itg2cnlem1  24296  isibl2  24301  iblcnlem  24323  limccnp2  24424  limcco  24425  dvmptres3  24487  itgsubstlem  24579  iblulm  24929  rlimcnp2  25477  dchrisumlema  25997  htthlem  28627  expgrowth  40551  mptelpm  41316  choicefi  41347  mullimc  41781  limcmptdm  41800  dvsinax  42081  dirkercncflem2  42274  fourierdlem62  42338  psmeasure  42638  ovnovollem2  42824  smflimsuplem2  42980
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