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Theorem dmmptdf2 39261
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 3210 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 450 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 2956 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 dmmptdf2.b . . . 4 𝑥𝐵
87rabid2f 3117 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
96, 8sylibr 224 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
10 dmmptdf2.a . . 3 𝐴 = (𝑥𝐵𝐶)
1110dmmpt 5628 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
129, 11syl6reqr 2674 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wnf 1707  wcel 1989  wnfc 2750  wral 2911  {crab 2915  Vcvv 3198  cmpt 4727  dom cdm 5112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by: (None)
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