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Theorem f0rn0 6782
Description: If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
Assertion
Ref Expression
f0rn0 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
Distinct variable groups:   𝑦,𝐸   𝑦,𝑌
Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem f0rn0
StepHypRef Expression
1 fdm 6731 . . 3 (𝐸:𝑋𝑌 → dom 𝐸 = 𝑋)
2 frn 6729 . . . . . . . . 9 (𝐸:𝑋𝑌 → ran 𝐸𝑌)
3 ralnex 3069 . . . . . . . . . 10 (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 ↔ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)
4 disj 4448 . . . . . . . . . . 11 ((𝑌 ∩ ran 𝐸) = ∅ ↔ ∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸)
5 df-ss 3964 . . . . . . . . . . . 12 (ran 𝐸𝑌 ↔ (ran 𝐸𝑌) = ran 𝐸)
6 incom 4201 . . . . . . . . . . . . . 14 (ran 𝐸𝑌) = (𝑌 ∩ ran 𝐸)
76eqeq1i 2733 . . . . . . . . . . . . 13 ((ran 𝐸𝑌) = ran 𝐸 ↔ (𝑌 ∩ ran 𝐸) = ran 𝐸)
8 eqtr2 2752 . . . . . . . . . . . . . 14 (((𝑌 ∩ ran 𝐸) = ran 𝐸 ∧ (𝑌 ∩ ran 𝐸) = ∅) → ran 𝐸 = ∅)
98ex 412 . . . . . . . . . . . . 13 ((𝑌 ∩ ran 𝐸) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
107, 9sylbi 216 . . . . . . . . . . . 12 ((ran 𝐸𝑌) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
115, 10sylbi 216 . . . . . . . . . . 11 (ran 𝐸𝑌 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
124, 11biimtrrid 242 . . . . . . . . . 10 (ran 𝐸𝑌 → (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
133, 12biimtrrid 242 . . . . . . . . 9 (ran 𝐸𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
142, 13syl 17 . . . . . . . 8 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
1514imp 406 . . . . . . 7 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → ran 𝐸 = ∅)
1615adantl 481 . . . . . 6 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → ran 𝐸 = ∅)
17 dm0rn0 5927 . . . . . 6 (dom 𝐸 = ∅ ↔ ran 𝐸 = ∅)
1816, 17sylibr 233 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → dom 𝐸 = ∅)
19 eqeq1 2732 . . . . . . 7 (𝑋 = dom 𝐸 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2019eqcoms 2736 . . . . . 6 (dom 𝐸 = 𝑋 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2120adantr 480 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2218, 21mpbird 257 . . . 4 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → 𝑋 = ∅)
2322exp32 420 . . 3 (dom 𝐸 = 𝑋 → (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅)))
241, 23mpcom 38 . 2 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅))
2524imp 406 1 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  wrex 3067  cin 3946  wss 3947  c0 4323  dom cdm 5678  ran crn 5679  wf 6544
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 5299  ax-nul 5306  ax-pr 5429
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-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5686  df-dm 5688  df-rn 5689  df-fn 6551  df-f 6552
This theorem is referenced by: (None)
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