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Theorem rnmpt0 39726
Description: The range of a function in map-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
rnmpt0.1 𝑥𝜑
rnmpt0.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmpt0.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmpt0 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rnmpt0
StepHypRef Expression
1 rnmpt0.1 . . . . . 6 𝑥𝜑
2 rnmpt0.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ex 449 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 2986 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 dmmptg 5670 . . . . 5 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
64, 5syl 17 . . . 4 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
76eqcomd 2657 . . 3 (𝜑𝐴 = dom (𝑥𝐴𝐵))
87eqeq1d 2653 . 2 (𝜑 → (𝐴 = ∅ ↔ dom (𝑥𝐴𝐵) = ∅))
9 dm0rn0 5374 . . 3 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
109a1i 11 . 2 (𝜑 → (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅))
11 rnmpt0.3 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
1211rneqi 5384 . . . . 5 ran 𝐹 = ran (𝑥𝐴𝐵)
1312a1i 11 . . . 4 (𝜑 → ran 𝐹 = ran (𝑥𝐴𝐵))
1413eqcomd 2657 . . 3 (𝜑 → ran (𝑥𝐴𝐵) = ran 𝐹)
1514eqeq1d 2653 . 2 (𝜑 → (ran (𝑥𝐴𝐵) = ∅ ↔ ran 𝐹 = ∅))
168, 10, 153bitrrd 295 1 (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941  c0 3948  cmpt 4762  dom cdm 5143  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  rnmptn0  39727
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