Theorem coeq0i 39357

Description: coeq0 6110 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 6522	. . . . . 6 ⊢ (𝐵:𝐸⟶𝐹 → ran 𝐵 ⊆ 𝐹)
2	1	3ad2ant2 1130	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → ran 𝐵 ⊆ 𝐹)
3		sslin 4213	. . . . 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 6524	. . . . . . 7 ⊢ (𝐴:𝐶⟶𝐷 → dom 𝐴 = 𝐶)
6	5	3ad2ant1 1129	. . . . . 6 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → dom 𝐴 = 𝐶)
7	6	ineq1d 4190	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = (𝐶 ∩ 𝐹))
8		simp3 1134	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐶 ∩ 𝐹) = ∅)
9	7, 8	eqtrd 2858	. . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = ∅)
10	4, 9	sseqtrd 4009	. . 3 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11		ss0 4354	. . 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 14341	1 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  dom cdm 5557  ran crn 5558   ∘ ccom 5561  ⟶wf 6353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-fn 6360  df-f 6361

This theorem is referenced by:  diophren  39417