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Theorem f0rn0 6794
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 6746 . . 3 (𝐸:𝑋𝑌 → dom 𝐸 = 𝑋)
2 frn 6744 . . . . . . . . 9 (𝐸:𝑋𝑌 → ran 𝐸𝑌)
3 ralnex 3070 . . . . . . . . . 10 (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 ↔ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)
4 disj 4456 . . . . . . . . . . 11 ((𝑌 ∩ ran 𝐸) = ∅ ↔ ∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸)
5 dfss2 3981 . . . . . . . . . . . 12 (ran 𝐸𝑌 ↔ (ran 𝐸𝑌) = ran 𝐸)
6 incom 4217 . . . . . . . . . . . . . 14 (ran 𝐸𝑌) = (𝑌 ∩ ran 𝐸)
76eqeq1i 2740 . . . . . . . . . . . . 13 ((ran 𝐸𝑌) = ran 𝐸 ↔ (𝑌 ∩ ran 𝐸) = ran 𝐸)
8 eqtr2 2759 . . . . . . . . . . . . . 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 217 . . . . . . . . . . . 12 ((ran 𝐸𝑌) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
115, 10sylbi 217 . . . . . . . . . . 11 (ran 𝐸𝑌 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
124, 11biimtrrid 243 . . . . . . . . . 10 (ran 𝐸𝑌 → (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
133, 12biimtrrid 243 . . . . . . . . 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 5938 . . . . . 6 (dom 𝐸 = ∅ ↔ ran 𝐸 = ∅)
1816, 17sylibr 234 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → dom 𝐸 = ∅)
19 eqeq1 2739 . . . . . . 7 (𝑋 = dom 𝐸 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2019eqcoms 2743 . . . . . 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  dom cdm 5689  ran crn 5690  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-f 6567
This theorem is referenced by: (None)
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