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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 2154 | . . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦 | |
2 | exsimpl 1869 | . . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧) | |
3 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelco 5744 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | breq2 5072 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧)) | |
7 | 6 | cbvexvw 2044 | . . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧) |
8 | 2, 5, 7 | 3imtr4i 294 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
9 | 1, 8 | exlimi 2217 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦) |
10 | 3 | eldm2 5772 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 ∘ 𝐵)) |
11 | 3 | eldm 5771 | . . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦) |
12 | 9, 10, 11 | 3imtr4i 294 | . 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵) |
13 | 12 | ssriv 3973 | 1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1780 ∈ wcel 2114 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 dom cdm 5557 ∘ ccom 5561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-co 5566 df-dm 5567 |
This theorem is referenced by: rncoss 5845 dmcosseq 5846 cossxp 6125 fvco4i 6764 cofunexg 7652 fin23lem30 9766 wunco 10157 relexpnndm 14402 mvdco 18575 f1omvdconj 18576 znleval 20703 ofco2 21062 tngtopn 23261 xppreima 30396 cycpmrn 30787 relexp0a 40068 dmtrclfvRP 40082 |
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