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Theorem rnmpt0 38921
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 450 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝑉))
41, 3ralrimi 2953 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
5 dmmptg 5601 . . . . 5 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
64, 5syl 17 . . . 4 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
76eqcomd 2627 . . 3 (𝜑𝐴 = dom (𝑥𝐴𝐵))
87eqeq1d 2623 . 2 (𝜑 → (𝐴 = ∅ ↔ dom (𝑥𝐴𝐵) = ∅))
9 dm0rn0 5312 . . 3 (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅)
109a1i 11 . 2 (𝜑 → (dom (𝑥𝐴𝐵) = ∅ ↔ ran (𝑥𝐴𝐵) = ∅))
11 rnmpt0.3 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
1211rneqi 5322 . . . . 5 ran 𝐹 = ran (𝑥𝐴𝐵)
1312a1i 11 . . . 4 (𝜑 → ran 𝐹 = ran (𝑥𝐴𝐵))
1413eqcomd 2627 . . 3 (𝜑 → ran (𝑥𝐴𝐵) = ran 𝐹)
1514eqeq1d 2623 . 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 384   = wceq 1480  wnf 1705  wcel 1987  wral 2908  c0 3897  cmpt 4683  dom cdm 5084  ran crn 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-mpt 4685  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by:  rnmptn0  38922
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