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| Mirrors > Home > MPE Home > Th. List > Mathboxes > coeq0i | Structured version Visualization version GIF version | ||
| Description: coeq0 6211 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 6666 | . . . . . 6 ⊢ (𝐵:𝐸⟶𝐹 → ran 𝐵 ⊆ 𝐹) | |
| 2 | 1 | 3ad2ant2 1141 | . . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → ran 𝐵 ⊆ 𝐹) |
| 3 | sslin 4174 | . . . . 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 6668 | . . . . . . 7 ⊢ (𝐴:𝐶⟶𝐷 → dom 𝐴 = 𝐶) | |
| 6 | 5 | 3ad2ant1 1140 | . . . . . 6 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → dom 𝐴 = 𝐶) |
| 7 | 6 | ineq1d 4151 | . . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = (𝐶 ∩ 𝐹)) |
| 8 | simp3 1145 | . . . . 5 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐶 ∩ 𝐹) = ∅) | |
| 9 | 7, 8 | eqtrd 2776 | . . . 4 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ 𝐹) = ∅) |
| 10 | 4, 9 | sseqtrd 3953 | . . 3 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (dom 𝐴 ∩ ran 𝐵) ⊆ ∅) |
| 11 | ss0 4333 | . . 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 14936 | 1 ⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1093 = wceq 1548 ∩ cin 3884 ⊆ wss 3885 ∅c0 4264 dom cdm 5621 ran crn 5622 ∘ ccom 5625 ⟶wf 6485 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-ext 2713 ax-sep 5221 ax-pr 5365 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-br 5076 df-opab 5138 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-fn 6492 df-f 6493 |
| This theorem is referenced by: diophren 43273 |
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