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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 7937 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 ∘ ccom 5682 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 |
This theorem is referenced by: rhmmpl 41786 |
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