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Mirrors > Home > MPE Home > Th. List > dmco | Structured version Visualization version GIF version |
Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Ref | Expression |
---|---|
dmco | ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 5728 | . 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵) | |
2 | cnvco 5720 | . . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
3 | 2 | rneqi 5771 | . 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴) |
4 | rnco2 6073 | . . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴) | |
5 | dfdm4 5728 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 5 | imaeq2i 5894 | . . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴) |
7 | 4, 6 | eqtr4i 2824 | . 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴) |
8 | 1, 3, 7 | 3eqtri 2825 | 1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 ∘ ccom 5523 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: curry1 7782 curry2 7785 smobeth 9997 hashkf 13688 imasless 16805 ofco2 21056 fcoinver 30370 xppreima 30408 smatrcl 31149 fco3 41857 |
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