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Theorem dmmptdf 39233
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
dmmptdf.x 𝑥𝜑
dmmptdf.a 𝐴 = (𝑥𝐵𝐶)
dmmptdf.c ((𝜑𝑥𝐵) → 𝐶𝑉)
Assertion
Ref Expression
dmmptdf (𝜑 → dom 𝐴 = 𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem dmmptdf
StepHypRef Expression
1 dmmptdf.x . . . 4 𝑥𝜑
2 dmmptdf.c . . . . . 6 ((𝜑𝑥𝐵) → 𝐶𝑉)
3 elex 3207 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 450 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 2954 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 rabid2 3113 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
86, 7sylibr 224 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
9 dmmptdf.a . . 3 𝐴 = (𝑥𝐵𝐶)
109dmmpt 5618 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
118, 10syl6reqr 2673 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wnf 1706  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  cmpt 4720  dom cdm 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-mpt 4721  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117
This theorem is referenced by:  smfpimltmpt  40718  smfpimltxrmpt  40730  smfadd  40736  smfpimgtmpt  40752  smfpimgtxrmpt  40755  smfpimioompt  40756  smfrec  40759  smfmul  40765  smfmulc1  40766  smffmpt  40774  smfsupmpt  40784  smfinfmpt  40788  smflimsupmpt  40798  smfliminfmpt  40801
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