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| Mirrors > Home > MPE Home > Th. List > coexd | Structured version Visualization version GIF version | ||
| Description: The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.) |
| Ref | Expression |
|---|---|
| coexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| coexd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| Ref | Expression |
|---|---|
| coexd | ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | coexd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | coexg 7951 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: rhmmpl 22387 rhmply1vr1 22391 rhmply1vsca 22392 1arithidom 33565 aks5lem2 42188 rhmpsr 42562 fuco112 49024 fuco111 49025 fuco112x 49027 |
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