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Theorem rnmpt0f 6242
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 3250 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 dmmptg 6241 . . . . 5 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
64, 5syl 17 . . . 4 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
76eqcomd 2734 . . 3 (𝜑𝐴 = dom (𝑥𝐴𝐵))
87eqeq1d 2730 . 2 (𝜑 → (𝐴 = ∅ ↔ dom (𝑥𝐴𝐵) = ∅))
9 dm0rn0 5922 . . 3 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
109a1i 11 . 2 (𝜑 → (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅))
11 rnmpt0f.3 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
1211rneqi 5934 . . . . 5 ran 𝐹 = ran (𝑥𝐴𝐵)
1312a1i 11 . . . 4 (𝜑 → ran 𝐹 = ran (𝑥𝐴𝐵))
1413eqcomd 2734 . . 3 (𝜑 → ran (𝑥𝐴𝐵) = ran 𝐹)
1514eqeq1d 2730 . 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 1534  wnf 1778  wcel 2099  wral 3057  c0 4319  cmpt 5226  dom cdm 5673  ran crn 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-mpt 5227  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686
This theorem is referenced by:  rnmptn0  6243
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