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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 2140 | . . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦 | |
2 | exsimpl 1864 | . . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧) | |
3 | vex 3466 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3466 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelco 5870 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | breq2 5149 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧)) | |
7 | 6 | cbvexvw 2033 | . . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧) |
8 | 2, 5, 7 | 3imtr4i 291 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
9 | 1, 8 | exlimi 2206 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
10 | 3 | eldm2 5900 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵)) |
11 | 3 | eldm 5899 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦) |
12 | 9, 10, 11 | 3imtr4i 291 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵) |
13 | 12 | ssriv 3982 | 1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∃wex 1774 ∈ wcel 2099 ⊆ wss 3946 〈cop 4629 class class class wbr 5145 dom cdm 5674 ∘ ccom 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-co 5683 df-dm 5684 |
This theorem is referenced by: rncoss 5971 dmcosseq 5972 cossxp 6275 fvco4i 6995 cofunexg 7954 fin23lem30 10376 wunco 10767 relexpnndm 15041 mvdco 19439 f1omvdconj 19440 znleval 21548 ofco2 22441 tngtopn 24655 xppreima 32563 cycpmrn 33025 relexp0a 43420 dmtrclfvRP 43434 |
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