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Mirrors > Home > ILE Home > Th. List > coss2 | GIF version |
Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.) |
Ref | Expression |
---|---|
coss2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | ssbrd 3971 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) |
3 | 2 | anim1d 334 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
4 | 3 | eximdv 1852 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
5 | 4 | ssopab2dv 4200 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}) |
6 | df-co 4548 | . 2 ⊢ (𝐶 ∘ 𝐴) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} | |
7 | df-co 4548 | . 2 ⊢ (𝐶 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)} | |
8 | 5, 6, 7 | 3sstr4g 3140 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1468 ⊆ wss 3071 class class class wbr 3929 {copab 3988 ∘ 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-in 3077 df-ss 3084 df-br 3930 df-opab 3990 df-co 4548 |
This theorem is referenced by: coeq2 4697 funss 5142 tposss 6143 dftpos4 6160 |
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