| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5187 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) | |
| 2 | 1 | anim2d 612 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))) |
| 3 | 2 | eximdv 1917 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))) |
| 4 | 3 | ssopab2dv 5556 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) |
| 5 | df-co 5694 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
| 6 | df-co 5694 | . 2 ⊢ (𝐵 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)} | |
| 7 | 4, 5, 6 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ⊆ wss 3951 class class class wbr 5143 {copab 5205 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 df-co 5694 |
| This theorem is referenced by: coeq1 5868 funss 6585 tposss 8252 cottrcl 9759 frmin 9789 frrlem16 9798 rtrclreclem4 15100 tsrdir 18649 ustex2sym 24225 ustex3sym 24226 ustneism 24232 trust 24238 utop2nei 24259 neipcfilu 24305 trclubgNEW 43631 trrelsuperrel2dg 43684 trclrelexplem 43724 cotrcltrcl 43738 cotrclrcl 43755 frege96d 43762 frege97d 43765 frege109d 43770 frege131d 43777 |
| Copyright terms: Public domain | W3C validator |