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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 2151 | . . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦 | |
2 | exsimpl 1869 | . . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧) | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelco 5706 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | breq2 5034 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧)) | |
7 | 6 | cbvexvw 2044 | . . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧) |
8 | 2, 5, 7 | 3imtr4i 295 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
9 | 1, 8 | exlimi 2215 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
10 | 3 | eldm2 5734 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵)) |
11 | 3 | eldm 5733 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦) |
12 | 9, 10, 11 | 3imtr4i 295 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵) |
13 | 12 | ssriv 3919 | 1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 class class class wbr 5030 dom cdm 5519 ∘ 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-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-co 5528 df-dm 5529 |
This theorem is referenced by: rncoss 5808 dmcosseq 5809 cossxp 6091 fvco4i 6739 cofunexg 7632 fin23lem30 9753 wunco 10144 relexpnndm 14392 mvdco 18565 f1omvdconj 18566 znleval 20246 ofco2 21056 tngtopn 23256 xppreima 30408 cycpmrn 30835 relexp0a 40417 dmtrclfvRP 40431 |
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