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Mirrors > Home > MPE Home > Th. List > coss1 | Structured version Visualization version GIF version |
Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.) |
Ref | Expression |
---|---|
coss1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbr 4729 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) | |
2 | 1 | anim2d 588 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))) |
3 | 2 | eximdv 1886 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))) |
4 | 3 | ssopab2dv 5033 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) |
5 | df-co 5152 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
6 | df-co 5152 | . 2 ⊢ (𝐵 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)} | |
7 | 4, 5, 6 | 3sstr4g 3679 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1744 ⊆ wss 3607 class class class wbr 4685 {copab 4745 ∘ ccom 5147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-in 3614 df-ss 3621 df-br 4686 df-opab 4746 df-co 5152 |
This theorem is referenced by: coeq1 5312 funss 5945 tposss 7398 rtrclreclem4 13845 tsrdir 17285 ustex2sym 22067 ustex3sym 22068 ustund 22072 ustneism 22074 trust 22080 utop2nei 22101 neipcfilu 22147 trclubgNEW 38242 trrelsuperrel2dg 38280 trclrelexplem 38320 cotrcltrcl 38334 cotrclrcl 38351 frege96d 38358 frege97d 38361 frege109d 38366 frege131d 38373 |
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