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Theorem f0rn0 6541
 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 6498 . . 3 (𝐸:𝑋𝑌 → dom 𝐸 = 𝑋)
2 frn 6496 . . . . . . . . 9 (𝐸:𝑋𝑌 → ran 𝐸𝑌)
3 ralnex 3199 . . . . . . . . . 10 (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 ↔ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)
4 disj 4355 . . . . . . . . . . 11 ((𝑌 ∩ ran 𝐸) = ∅ ↔ ∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸)
5 df-ss 3898 . . . . . . . . . . . 12 (ran 𝐸𝑌 ↔ (ran 𝐸𝑌) = ran 𝐸)
6 incom 4128 . . . . . . . . . . . . . 14 (ran 𝐸𝑌) = (𝑌 ∩ ran 𝐸)
76eqeq1i 2803 . . . . . . . . . . . . 13 ((ran 𝐸𝑌) = ran 𝐸 ↔ (𝑌 ∩ ran 𝐸) = ran 𝐸)
8 eqtr2 2819 . . . . . . . . . . . . . 14 (((𝑌 ∩ ran 𝐸) = ran 𝐸 ∧ (𝑌 ∩ ran 𝐸) = ∅) → ran 𝐸 = ∅)
98ex 416 . . . . . . . . . . . . 13 ((𝑌 ∩ ran 𝐸) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
107, 9sylbi 220 . . . . . . . . . . . 12 ((ran 𝐸𝑌) = ran 𝐸 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
115, 10sylbi 220 . . . . . . . . . . 11 (ran 𝐸𝑌 → ((𝑌 ∩ ran 𝐸) = ∅ → ran 𝐸 = ∅))
124, 11syl5bir 246 . . . . . . . . . 10 (ran 𝐸𝑌 → (∀𝑦𝑌 ¬ 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
133, 12syl5bir 246 . . . . . . . . 9 (ran 𝐸𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
142, 13syl 17 . . . . . . . 8 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸 → ran 𝐸 = ∅))
1514imp 410 . . . . . . 7 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → ran 𝐸 = ∅)
1615adantl 485 . . . . . 6 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → ran 𝐸 = ∅)
17 dm0rn0 5760 . . . . . 6 (dom 𝐸 = ∅ ↔ ran 𝐸 = ∅)
1816, 17sylibr 237 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → dom 𝐸 = ∅)
19 eqeq1 2802 . . . . . . 7 (𝑋 = dom 𝐸 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2019eqcoms 2806 . . . . . 6 (dom 𝐸 = 𝑋 → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2120adantr 484 . . . . 5 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → (𝑋 = ∅ ↔ dom 𝐸 = ∅))
2218, 21mpbird 260 . . . 4 ((dom 𝐸 = 𝑋 ∧ (𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸)) → 𝑋 = ∅)
2322exp32 424 . . 3 (dom 𝐸 = 𝑋 → (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅)))
241, 23mpcom 38 . 2 (𝐸:𝑋𝑌 → (¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸𝑋 = ∅))
2524imp 410 1 ((𝐸:𝑋𝑌 ∧ ¬ ∃𝑦𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  dom cdm 5520  ran crn 5521  ⟶wf 6323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-cnv 5528  df-dm 5530  df-rn 5531  df-fn 6330  df-f 6331 This theorem is referenced by: (None)
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