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Theorem f0rn0 6128
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 6089 . . 3 (𝐸:𝑋𝑌 → dom 𝐸 = 𝑋)
2 frn 6091 . . . . . . . . 9 (𝐸:𝑋𝑌 → ran 𝐸𝑌)
3 ralnex 3021 . . . . . . . . . 10 (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 ↔ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)
4 disj 4050 . . . . . . . . . . 11 ((𝑌 ∩ ran 𝐸) = ∅ ↔ ∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸)
5 df-ss 3621 . . . . . . . . . . . 12 (ran 𝐸𝑌 ↔ (ran 𝐸𝑌) = ran 𝐸)
6 incom 3838 . . . . . . . . . . . . . 14 (ran 𝐸𝑌) = (𝑌 ∩ ran 𝐸)
76eqeq1i 2656 . . . . . . . . . . . . 13 ((ran 𝐸𝑌) = ran 𝐸 ↔ (𝑌 ∩ ran 𝐸) = ran 𝐸)
8 eqtr2 2671 . . . . . . . . . . . . . 14 (((𝑌 ∩ ran 𝐸) = ran 𝐸 ∧ (𝑌 ∩ ran 𝐸) = ∅) → ran 𝐸 = ∅)
98ex 449 . . . . . . . . . . . . 13 ((𝑌 ∩ ran 𝐸) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
107, 9sylbi 207 . . . . . . . . . . . 12 ((ran 𝐸𝑌) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
115, 10sylbi 207 . . . . . . . . . . 11 (ran 𝐸𝑌 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
124, 11syl5bir 233 . . . . . . . . . 10 (ran 𝐸𝑌 → (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
133, 12syl5bir 233 . . . . . . . . 9 (ran 𝐸𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
142, 13syl 17 . . . . . . . 8 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
1514imp 444 . . . . . . 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 5374 . . . . . 6 (dom 𝐸 = ∅ ↔ ran 𝐸 = ∅)
1816, 17sylibr 224 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → dom 𝐸 = ∅)
19 eqeq1 2655 . . . . . . 7 (𝑋 = dom 𝐸 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2019eqcoms 2659 . . . . . 6 (dom 𝐸 = 𝑋 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2120adantr 480 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2218, 21mpbird 247 . . . 4 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → 𝑋 = ∅)
2322exp32 630 . . 3 (dom 𝐸 = 𝑋 → (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅)))
241, 23mpcom 38 . 2 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅))
2524imp 444 1 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948  dom cdm 5143  ran crn 5144  wf 5922
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-rex 2947  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-cnv 5151  df-dm 5153  df-rn 5154  df-fn 5929  df-f 5930
This theorem is referenced by: (None)
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