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| Mirrors > Home > MPE Home > Th. List > dmcoeq | Structured version Visualization version GIF version | ||
| Description: Domain of a composition. (Contributed by NM, 19-Mar-1998.) |
| Ref | Expression |
|---|---|
| dmcoeq | ⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqimss2 4043 | . 2 ⊢ (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴) | |
| 2 | dmcosseq 5987 | . 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 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-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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: rncoeq 5990 dfdm2 6301 funcocnv2 6873 |
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