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Theorem dmmptdf2 41794
 Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
dmmptdf2.x 𝑥𝜑
dmmptdf2.b 𝑥𝐵
dmmptdf2.a 𝐴 = (𝑥𝐵𝐶)
dmmptdf2.c ((𝜑𝑥𝐵) → 𝐶𝑉)
Assertion
Ref Expression
dmmptdf2 (𝜑 → dom 𝐴 = 𝐵)

Proof of Theorem dmmptdf2
StepHypRef Expression
1 dmmptdf2.x . . . 4 𝑥𝜑
2 dmmptdf2.c . . . . 5 ((𝜑𝑥𝐵) → 𝐶𝑉)
32elexd 3500 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
41, 3ralrimia 41689 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
5 dmmptdf2.b . . . 4 𝑥𝐵
65rabid2f 3373 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
74, 6sylibr 237 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
8 dmmptdf2.a . . 3 𝐴 = (𝑥𝐵𝐶)
98dmmpt 6081 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
107, 9syl6reqr 2878 1 (𝜑 → dom 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2115  Ⅎwnfc 2962  ∀wral 3133  {crab 3137  Vcvv 3480   ↦ cmpt 5132  dom cdm 5542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-mpt 5133  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555 This theorem is referenced by: (None)
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