Theorem coeq0i 43339

Description: coeq0 6245 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Assertion

Ref	Expression
coeq0i	⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅)

Proof of Theorem coeq0i

Step	Hyp	Ref	Expression
1		frn 6701	. . . . . 6 ⊢ (𝐵:𝐸⟶𝐹 → ran 𝐵 ⊆ 𝐹)
2	1	3ad2ant2 1148	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → ran 𝐵 ⊆ 𝐹)
3		sslin 4196	. . . . 5 ⊢ (ran 𝐵 ⊆ 𝐹 → (dom 𝐴 ∩ ran 𝐵) ⊆ (dom 𝐴 ∩ 𝐹))
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ (dom 𝐴 ∩ 𝐹))
5		fdm 6703	. . . . . . 7 ⊢ (𝐴:𝐶⟶𝐷 → dom 𝐴 = 𝐶)
6	5	3ad2ant1 1147	. . . . . 6 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → dom 𝐴 = 𝐶)
7	6	ineq1d 4173	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = (𝐶 ∩ 𝐹))
8		simp3 1152	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐶 ∩ 𝐹) = ∅)
9	7, 8	eqtrd 2799	. . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = ∅)
10	4, 9	sseqtrd 3974	. . 3 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11		ss0 4358	. . 3 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ ∅ → (dom 𝐴 ∩ ran 𝐵) = ∅)
12	10, 11	syl 17	. 2 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) = ∅)
13	12	coemptyd 14994	1 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  dom cdm 5649  ran crn 5650   ∘ ccom 5653  ⟶wf 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-fn 6526  df-f 6527

This theorem is referenced by:  diophren  43395