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| 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 2149 | . . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦 | |
| 2 | exsimpl 1867 | . . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧) | |
| 3 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 4 | vex 3483 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | opelco 5881 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | 
| 6 | breq2 5146 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧)) | |
| 7 | 6 | cbvexvw 2035 | . . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧) | 
| 8 | 2, 5, 7 | 3imtr4i 292 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) | 
| 9 | 1, 8 | exlimi 2216 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) | 
| 10 | 3 | eldm2 5911 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵)) | 
| 11 | 3 | eldm 5910 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦) | 
| 12 | 9, 10, 11 | 3imtr4i 292 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵) | 
| 13 | 12 | ssriv 3986 | 1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∃wex 1778 ∈ wcel 2107 ⊆ wss 3950 〈cop 4631 class class class wbr 5142 dom cdm 5684 ∘ ccom 5688 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-co 5693 df-dm 5694 | 
| This theorem is referenced by: rncoss 5985 dmcosseq 5986 dmcosseqOLD 5987 cossxp 6291 fvco4i 7009 cofunexg 7974 fin23lem30 10383 wunco 10774 relexpnndm 15081 mvdco 19464 f1omvdconj 19465 znleval 21574 ofco2 22458 tngtopn 24672 xppreima 32656 cycpmrn 33164 relexp0a 43734 dmtrclfvRP 43748 dmtposss 48782 | 
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