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Theorem rnmpt0f 6233
Description: The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
rnmpt0f.1 𝑥𝜑
rnmpt0f.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmpt0f.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmpt0f (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rnmpt0f
StepHypRef Expression
1 rnmpt0f.1 . . . . . 6 𝑥𝜑
2 rnmpt0f.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ex 412 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 3246 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 dmmptg 6232 . . . . 5 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
64, 5syl 17 . . . 4 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
76eqcomd 2730 . . 3 (𝜑𝐴 = dom (𝑥𝐴𝐵))
87eqeq1d 2726 . 2 (𝜑 → (𝐴 = ∅ ↔ dom (𝑥𝐴𝐵) = ∅))
9 dm0rn0 5915 . . 3 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
109a1i 11 . 2 (𝜑 → (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅))
11 rnmpt0f.3 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
1211rneqi 5927 . . . . 5 ran 𝐹 = ran (𝑥𝐴𝐵)
1312a1i 11 . . . 4 (𝜑 → ran 𝐹 = ran (𝑥𝐴𝐵))
1413eqcomd 2730 . . 3 (𝜑 → ran (𝑥𝐴𝐵) = ran 𝐹)
1514eqeq1d 2726 . 2 (𝜑 → (ran (𝑥𝐴𝐵) = ∅ ↔ ran 𝐹 = ∅))
168, 10, 153bitrrd 306 1 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wnf 1777  wcel 2098  wral 3053  c0 4315  cmpt 5222  dom cdm 5667  ran crn 5668
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-mpt 5223  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680
This theorem is referenced by:  rnmptn0  6234
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