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Theorem dmmptdf2 39907
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 3340 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 449 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 3083 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 dmmptdf2.b . . . 4 𝑥𝐵
87rabid2f 3246 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
96, 8sylibr 224 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
10 dmmptdf2.a . . 3 𝐴 = (𝑥𝐵𝐶)
1110dmmpt 5779 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
129, 11syl6reqr 2801 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wnf 1845  wcel 2127  wnfc 2877  wral 3038  {crab 3042  Vcvv 3328  cmpt 4869  dom cdm 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-mpt 4870  df-xp 5260  df-rel 5261  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267
This theorem is referenced by: (None)
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