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Theorem coeq0i 36334
 Description: coeq0 5561 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 5966 . . . . . 6 (𝐵:𝐸𝐹 → ran 𝐵𝐹)
213ad2ant2 1076 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → ran 𝐵𝐹)
3 sslin 3801 . . . . 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 5964 . . . . . . 7 (𝐴:𝐶𝐷 → dom 𝐴 = 𝐶)
653ad2ant1 1075 . . . . . 6 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → dom 𝐴 = 𝐶)
76ineq1d 3775 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = (𝐶𝐹))
8 simp3 1056 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐶𝐹) = ∅)
97, 8eqtrd 2644 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = ∅)
104, 9sseqtrd 3604 . . 3 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11 ss0 3926 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
1210, 11syl 17 . 2 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
13 coeq0 5561 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
1412, 13sylibr 223 1 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fn 5807  df-f 5808 This theorem is referenced by:  diophren  36395
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