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

Proof of Theorem coeq0i

Step	Hyp	Ref	Expression
1		frn 6727	. . . . . 6 ⊢ (𝐵:𝐸⟶𝐹 → ran 𝐵 ⊆ 𝐹)
2	1	3ad2ant2 1131	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → ran 𝐵 ⊆ 𝐹)
3		sslin 4233	. . . . 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 6729	. . . . . . 7 ⊢ (𝐴:𝐶⟶𝐷 → dom 𝐴 = 𝐶)
6	5	3ad2ant1 1130	. . . . . 6 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → dom 𝐴 = 𝐶)
7	6	ineq1d 4209	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = (𝐶 ∩ 𝐹))
8		simp3 1135	. . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐶 ∩ 𝐹) = ∅)
9	7, 8	eqtrd 2766	. . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = ∅)
10	4, 9	sseqtrd 4019	. . 3 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅)
11		ss0 4396	. . 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 14979	1 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅)

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