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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 5893 | . 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵) | |
2 | cnvco 5883 | . . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
3 | 2 | rneqi 5934 | . 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴) |
4 | rnco2 6249 | . . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴) | |
5 | dfdm4 5893 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 5 | imaeq2i 6055 | . . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴) |
7 | 4, 6 | eqtr4i 2763 | . 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴) |
8 | 1, 3, 7 | 3eqtri 2764 | 1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ◡ccnv 5674 dom cdm 5675 ran crn 5676 “ cima 5678 ∘ ccom 5679 |
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-11 2154 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: fncofn 6663 fco3OLD 6748 curry1 8086 curry2 8089 smobeth 10577 hashkf 14288 imasless 17482 ofco2 21944 fcoinver 31822 xppreima 31858 smatrcl 32764 |
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