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Theorem coeq0i 36224
Description: coeq0 5451 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
coeq0i ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)

Proof of Theorem coeq0i
StepHypRef Expression
1 frn 5851 . . . . . 6 (𝐵:𝐸𝐹 → ran 𝐵𝐹)
213ad2ant2 1075 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → ran 𝐵𝐹)
3 sslin 3704 . . . . 5 (ran 𝐵𝐹 → (dom 𝐴 ∩ ran 𝐵) ⊆ (dom 𝐴𝐹))
42, 3syl 17 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ (dom 𝐴𝐹))
5 fdm 5849 . . . . . . 7 (𝐴:𝐶𝐷 → dom 𝐴 = 𝐶)
653ad2ant1 1074 . . . . . 6 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → dom 𝐴 = 𝐶)
76ineq1d 3678 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = (𝐶𝐹))
8 simp3 1055 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐶𝐹) = ∅)
97, 8eqtrd 2548 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = ∅)
104, 9sseqtrd 3508 . . 3 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11 ss0 3829 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
1210, 11syl 17 . 2 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
13 coeq0 5451 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
1412, 13sylibr 222 1 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  cin 3443  wss 3444  c0 3777  dom cdm 4932  ran crn 4933  ccom 4936  wf 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-fn 5692  df-f 5693
This theorem is referenced by:  diophren  36285
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