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Mirrors > Home > MPE Home > Th. List > dmcoss | Structured version Visualization version GIF version |
Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmcoss | ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2150 | . . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦 | |
2 | exsimpl 1865 | . . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧) | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelco 5736 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | breq2 5062 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧)) | |
7 | 6 | cbvexvw 2040 | . . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧) |
8 | 2, 5, 7 | 3imtr4i 294 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
9 | 1, 8 | exlimi 2213 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
10 | 3 | eldm2 5764 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵)) |
11 | 3 | eldm 5763 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦) |
12 | 9, 10, 11 | 3imtr4i 294 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵) |
13 | 12 | ssriv 3970 | 1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ⊆ wss 3935 〈cop 4566 class class class wbr 5058 dom cdm 5549 ∘ ccom 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-co 5558 df-dm 5559 |
This theorem is referenced by: rncoss 5837 dmcosseq 5838 cossxp 6117 fvco4i 6756 cofunexg 7644 fin23lem30 9758 wunco 10149 relexpnndm 14394 mvdco 18567 f1omvdconj 18568 znleval 20695 ofco2 21054 tngtopn 23253 xppreima 30388 cycpmrn 30780 relexp0a 40054 dmtrclfvRP 40068 |
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