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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 4899 | . 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) | |
2 | eqcom 2179 | . . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴) | |
3 | df-rn 4637 | . . . 4 ⊢ ran 𝐵 = dom ◡𝐵 | |
4 | dfdm4 4819 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
5 | 3, 4 | eqeq12i 2191 | . . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴) |
6 | 2, 5 | bitri 184 | . 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴) |
7 | df-rn 4637 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
8 | cnvco 4812 | . . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 8 | dmeqi 4828 | . . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
10 | 7, 9 | eqtri 2198 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
11 | df-rn 4637 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
12 | 10, 11 | eqeq12i 2191 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) |
13 | 1, 6, 12 | 3imtr4i 201 | 1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ◡ccnv 4625 dom cdm 4626 ran crn 4627 ∘ ccom 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 |
This theorem is referenced by: dfdm2 5163 foco 5448 |
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