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Theorem dmmptdf2 40187
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 . . . . . 6 ((𝜑𝑥𝐵) → 𝐶𝑉)
3 elex 3400 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 402 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 3138 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 dmmptdf2.b . . . 4 𝑥𝐵
87rabid2f 3301 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
96, 8sylibr 226 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
10 dmmptdf2.a . . 3 𝐴 = (𝑥𝐵𝐶)
1110dmmpt 5849 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
129, 11syl6reqr 2852 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wnf 1879  wcel 2157  wnfc 2928  wral 3089  {crab 3093  Vcvv 3385  cmpt 4922  dom cdm 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-mpt 4923  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325
This theorem is referenced by: (None)
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