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Mirrors > Home > ILE Home > Th. List > coexg | GIF version |
Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Ref | Expression |
---|---|
coexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossxp 5133 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | |
2 | dmexg 4875 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom 𝐵 ∈ V) | |
3 | rnexg 4876 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
4 | xpexg 4725 | . . 3 ⊢ ((dom 𝐵 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐵 × ran 𝐴) ∈ V) | |
5 | 2, 3, 4 | syl2anr 288 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (dom 𝐵 × ran 𝐴) ∈ V) |
6 | ssexg 4128 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 412 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 × cxp 4609 dom cdm 4611 ran crn 4612 ∘ ccom 4615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 |
This theorem is referenced by: coex 5156 climcncf 13365 |
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