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Theorem coeq0i 42447
Description: coeq0 6258 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 6727 . . . . . 6 (𝐵:𝐸𝐹 → ran 𝐵𝐹)
213ad2ant2 1131 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → ran 𝐵𝐹)
3 sslin 4233 . . . . 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 6729 . . . . . . 7 (𝐴:𝐶𝐷 → dom 𝐴 = 𝐶)
653ad2ant1 1130 . . . . . 6 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → dom 𝐴 = 𝐶)
76ineq1d 4209 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = (𝐶𝐹))
8 simp3 1135 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐶𝐹) = ∅)
97, 8eqtrd 2766 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = ∅)
104, 9sseqtrd 4019 . . 3 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11 ss0 4396 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
1210, 11syl 17 . 2 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
1312coemptyd 14979 1 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  cin 3945  wss 3946  c0 4322  dom cdm 5674  ran crn 5675  ccom 5678  wf 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-fn 6549  df-f 6550
This theorem is referenced by:  diophren  42507
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