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Theorem dmmptdf 40158
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 3399 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 402 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 3137 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 rabid2 3299 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
86, 7sylibr 226 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
9 dmmptdf.a . . 3 𝐴 = (𝑥𝐵𝐶)
109dmmpt 5848 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
118, 10syl6reqr 2851 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wnf 1879  wcel 2157  wral 3088  {crab 3092  Vcvv 3384  cmpt 4921  dom cdm 5311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4843  df-opab 4905  df-mpt 4922  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324
This theorem is referenced by:  smfpimltmpt  41690  smfpimltxrmpt  41702  smfadd  41708  smfpimgtmpt  41724  smfpimgtxrmpt  41727  smfpimioompt  41728  smfrec  41731  smfmul  41737  smfmulc1  41738  smffmpt  41746  smfsupmpt  41756  smfinfmpt  41760  smflimsupmpt  41770  smfliminfmpt  41773
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