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Theorem coeq0i 38286
Description: coeq0 5900 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 6299 . . . . . 6 (𝐵:𝐸𝐹 → ran 𝐵𝐹)
213ad2ant2 1125 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → ran 𝐵𝐹)
3 sslin 4059 . . . . 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 6301 . . . . . . 7 (𝐴:𝐶𝐷 → dom 𝐴 = 𝐶)
653ad2ant1 1124 . . . . . 6 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → dom 𝐴 = 𝐶)
76ineq1d 4036 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = (𝐶𝐹))
8 simp3 1129 . . . . 5 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐶𝐹) = ∅)
97, 8eqtrd 2814 . . . 4 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴𝐹) = ∅)
104, 9sseqtrd 3860 . . 3 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11 ss0 4200 . . 3 ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
1210, 11syl 17 . 2 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
13 coeq0 5900 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
1412, 13sylibr 226 1 ((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1601  cin 3791  wss 3792  c0 4141  dom cdm 5357  ran crn 5358  ccom 5361  wf 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-fn 6140  df-f 6141
This theorem is referenced by:  diophren  38347
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