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Theorem dmmptdf 44733
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 2891 . 2 𝑥𝐵
3 dmmptdf.a . 2 𝐴 = (𝑥𝐵𝐶)
4 dmmptdf.c . 2 ((𝜑𝑥𝐵) → 𝐶𝑉)
51, 2, 3, 4dmmptdff 44732 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  cmpt 5232  dom cdm 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691
This theorem is referenced by:  smfpimltmpt  46269  smfadd  46288  smfpimgtmpt  46304  smfpimioompt  46309  smfrec  46312  smfmul  46318  smfmulc1  46319  smfsupmpt  46338  smfinfmpt  46342  smflimsupmpt  46352  smfliminfmpt  46355
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