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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 3979 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) |
3 | 2 | anim1d 334 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
4 | 3 | eximdv 1853 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
5 | 4 | ssopab2dv 4208 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}) |
6 | df-co 4556 | . 2 ⊢ (𝐶 ∘ 𝐴) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} | |
7 | df-co 4556 | . 2 ⊢ (𝐶 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)} | |
8 | 5, 6, 7 | 3sstr4g 3145 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1469 ⊆ wss 3076 class class class wbr 3937 {copab 3996 ∘ ccom 4551 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-in 3082 df-ss 3089 df-br 3938 df-opab 3998 df-co 4556 |
This theorem is referenced by: coeq2 4705 funss 5150 tposss 6151 dftpos4 6168 |
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