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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 5061 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) | |
2 | dmexg 4803 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom 𝐵 ∈ V) | |
3 | rnexg 4804 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
4 | xpexg 4653 | . . 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 4067 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 410 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 × cxp 4537 dom cdm 4539 ran crn 4540 ∘ ccom 4543 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 |
This theorem is referenced by: coex 5084 climcncf 12740 |
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