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Theorem coeq0i 37836
Description: coeq0 5805 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 6214 . . . . . 6 (𝐵:𝐸𝐹 → ran 𝐵𝐹)
213ad2ant2 1129 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → ran 𝐵𝐹)
3 sslin 3982 . . . . 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 6212 . . . . . . 7 (𝐴:𝐶𝐷 → dom 𝐴 = 𝐶)
653ad2ant1 1128 . . . . . 6 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → dom 𝐴 = 𝐶)
76ineq1d 3956 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = (𝐶𝐹))
8 simp3 1133 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐶𝐹) = ∅)
97, 8eqtrd 2794 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = ∅)
104, 9sseqtrd 3782 . . 3 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11 ss0 4117 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
1210, 11syl 17 . 2 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
13 coeq0 5805 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
1412, 13sylibr 224 1 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  cin 3714  wss 3715  c0 4058  dom cdm 5266  ran crn 5267  ccom 5270  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-fn 6052  df-f 6053
This theorem is referenced by:  diophren  37897
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