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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 5929 | . 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) | |
2 | eqcom 2743 | . . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴) | |
3 | df-rn 5644 | . . . 4 ⊢ ran 𝐵 = dom ◡𝐵 | |
4 | dfdm4 5851 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
5 | 3, 4 | eqeq12i 2754 | . . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴) |
6 | 2, 5 | bitri 274 | . 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴) |
7 | df-rn 5644 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
8 | cnvco 5841 | . . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 8 | dmeqi 5860 | . . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
10 | 7, 9 | eqtri 2764 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
11 | df-rn 5644 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
12 | 10, 11 | eqeq12i 2754 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴) |
13 | 1, 6, 12 | 3imtr4i 291 | 1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ◡ccnv 5632 dom cdm 5633 ran crn 5634 ∘ ccom 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 |
This theorem is referenced by: dfdm2 6233 |
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