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Theorem dmmptdf 45229
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 . 2 𝑥𝜑
2 nfcv 2905 . 2 𝑥𝐵
3 dmmptdf.a . 2 𝐴 = (𝑥𝐵𝐶)
4 dmmptdf.c . 2 ((𝜑𝑥𝐵) → 𝐶𝑉)
51, 2, 3, 4dmmptdff 45228 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  cmpt 5225  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  smfpimltmpt  46761  smfadd  46780  smfpimgtmpt  46796  smfpimioompt  46801  smfrec  46804  smfmul  46810  smfmulc1  46811  smfsupmpt  46830  smfinfmpt  46834  smflimsupmpt  46844  smfliminfmpt  46847
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