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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 3796 | . . . . 5 ⊢ (A ⊆ B → (xAy → xBy)) |
3 | 2 | anim1d 319 | . . . 4 ⊢ (A ⊆ B → ((xAy ∧ y𝐶z) → (xBy ∧ y𝐶z))) |
4 | 3 | eximdv 1757 | . . 3 ⊢ (A ⊆ B → (∃y(xAy ∧ y𝐶z) → ∃y(xBy ∧ y𝐶z))) |
5 | 4 | ssopab2dv 4006 | . 2 ⊢ (A ⊆ B → {〈x, z〉 ∣ ∃y(xAy ∧ y𝐶z)} ⊆ {〈x, z〉 ∣ ∃y(xBy ∧ y𝐶z)}) |
6 | df-co 4297 | . 2 ⊢ (𝐶 ∘ A) = {〈x, z〉 ∣ ∃y(xAy ∧ y𝐶z)} | |
7 | df-co 4297 | . 2 ⊢ (𝐶 ∘ B) = {〈x, z〉 ∣ ∃y(xBy ∧ y𝐶z)} | |
8 | 5, 6, 7 | 3sstr4g 2980 | 1 ⊢ (A ⊆ B → (𝐶 ∘ A) ⊆ (𝐶 ∘ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ⊆ wss 2911 class class class wbr 3755 {copab 3808 ∘ ccom 4292 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-in 2918 df-ss 2925 df-br 3756 df-opab 3810 df-co 4297 |
This theorem is referenced by: coeq2 4437 funss 4863 tposss 5802 dftpos4 5819 |
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