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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 5238 | . 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵) | |
2 | cnvco 5230 | . . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
3 | 2 | rneqi 5273 | . 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴) |
4 | rnco2 5559 | . . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴) | |
5 | dfdm4 5238 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | 5 | imaeq2i 5383 | . . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴) |
7 | 4, 6 | eqtr4i 2635 | . 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴) |
8 | 1, 3, 7 | 3eqtri 2636 | 1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: curry1 7156 curry2 7159 smobeth 9287 hashkf 12981 imasless 16023 ofco2 20076 fcoinver 28798 xppreima 28829 smatrcl 29190 fco3 38416 |
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