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Mirrors > Home > ILE Home > Th. List > rncoeq | 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 4870 | . 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) | |
2 | eqcom 2166 | . . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴) | |
3 | df-rn 4609 | . . . 4 ⊢ ran 𝐵 = dom ◡𝐵 | |
4 | dfdm4 4790 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
5 | 3, 4 | eqeq12i 2178 | . . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴) |
6 | 2, 5 | bitri 183 | . 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴) |
7 | df-rn 4609 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
8 | cnvco 4783 | . . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 8 | dmeqi 4799 | . . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
10 | 7, 9 | eqtri 2185 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
11 | df-rn 4609 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
12 | 10, 11 | eqeq12i 2178 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) |
13 | 1, 6, 12 | 3imtr4i 200 | 1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ◡ccnv 4597 dom cdm 4598 ran crn 4599 ∘ ccom 4602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 |
This theorem is referenced by: dfdm2 5132 foco 5414 |
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