Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5757 | . 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵) | |
2 | cnvco 5749 | . . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
3 | 2 | rneqi 5800 | . 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴) |
4 | rnco2 6099 | . . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴) | |
5 | dfdm4 5757 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 5 | imaeq2i 5920 | . . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴) |
7 | 4, 6 | eqtr4i 2844 | . 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴) |
8 | 1, 3, 7 | 3eqtri 2845 | 1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: curry1 7788 curry2 7791 smobeth 9996 hashkf 13680 imasless 16801 ofco2 20988 fcoinver 30285 xppreima 30322 smatrcl 30960 fco3 41367 |
Copyright terms: Public domain | W3C validator |