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Theorem dmmptdf 39935
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 3353 . . . . . 6 (𝐶𝑉𝐶 ∈ V)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
54ex 449 . . . 4 (𝜑 → (𝑥𝐵𝐶 ∈ V))
61, 5ralrimi 3096 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 rabid2 3258 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
86, 7sylibr 224 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
9 dmmptdf.a . . 3 𝐴 = (𝑥𝐵𝐶)
109dmmpt 5792 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
118, 10syl6reqr 2814 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wnf 1857  wcel 2140  wral 3051  {crab 3055  Vcvv 3341  cmpt 4882  dom cdm 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-mpt 4883  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280
This theorem is referenced by:  smfpimltmpt  41480  smfpimltxrmpt  41492  smfadd  41498  smfpimgtmpt  41514  smfpimgtxrmpt  41517  smfpimioompt  41518  smfrec  41521  smfmul  41527  smfmulc1  41528  smffmpt  41536  smfsupmpt  41546  smfinfmpt  41550  smflimsupmpt  41560  smfliminfmpt  41563
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