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| Mirrors > Home > MPE Home > Th. List > rncoeq | Structured version Visualization version GIF version | ||
| Description: Range of a composition. (Contributed by NM, 19-Mar-1998.) |
| Ref | Expression |
|---|---|
| rncoeq | ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmcoeq 5989 | . 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) | |
| 2 | eqcom 2744 | . . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴) | |
| 3 | df-rn 5696 | . . . 4 ⊢ ran 𝐵 = dom ◡𝐵 | |
| 4 | dfdm4 5906 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
| 5 | 3, 4 | eqeq12i 2755 | . . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴) |
| 6 | 2, 5 | bitri 275 | . 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴) |
| 7 | df-rn 5696 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
| 8 | cnvco 5896 | . . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
| 9 | 8 | dmeqi 5915 | . . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
| 10 | 7, 9 | eqtri 2765 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
| 11 | df-rn 5696 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
| 12 | 10, 11 | eqeq12i 2755 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) |
| 13 | 1, 6, 12 | 3imtr4i 292 | 1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ◡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-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: dfdm2 6301 algextdeglem4 33761 |
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