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Mirrors > Home > MPE Home > Th. List > Mathboxes > coeq0i | Structured version Visualization version GIF version |
Description: coeq0 6075 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
coeq0i | ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frn 6493 | . . . . . 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 4161 | . . . . 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 6495 | . . . . . . 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 4138 | . . . . 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 2833 | . . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = ∅) |
10 | 4, 9 | sseqtrd 3955 | . . 3 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅) |
11 | ss0 4306 | . . 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 14330 | 1 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 dom cdm 5519 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-fn 6327 df-f 6328 |
This theorem is referenced by: diophren 39754 |
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