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| Mirrors > Home > MPE Home > Th. List > Mathboxes > conrel1d | Structured version Visualization version GIF version | ||
| Description: Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
| Ref | Expression |
|---|---|
| conrel1d.a | ⊢ (𝜑 → ◡𝐴 = ∅) |
| Ref | Expression |
|---|---|
| conrel1d | ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | incom 4209 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
| 2 | dfdm4 5906 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
| 3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
| 4 | 3 | rneqd 5949 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
| 5 | rn0 5936 | . . . . . 6 ⊢ ran ∅ = ∅ | |
| 6 | 4, 5 | eqtrdi 2793 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
| 7 | 2, 6 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
| 8 | ineq2 4214 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
| 9 | in0 4395 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
| 10 | 8, 9 | eqtrdi 2793 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
| 11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
| 12 | 1, 11 | eqtrid 2789 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
| 13 | 12 | coemptyd 15018 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ∅c0 4333 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 |
| This theorem is referenced by: (None) |
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