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| Mirrors > Home > MPE Home > Th. List > coss12d | Structured version Visualization version GIF version | ||
| Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.) |
| Ref | Expression |
|---|---|
| coss12d.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| coss12d.c | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
| Ref | Expression |
|---|---|
| coss12d | ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coss12d.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) | |
| 2 | 1 | ssbrd 5186 | . . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦)) |
| 3 | coss12d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 4 | 3 | ssbrd 5186 | . . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) |
| 5 | 2, 4 | anim12d 609 | . . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
| 6 | 5 | eximdv 1917 | . . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
| 7 | 6 | ssopab2dv 5556 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}) |
| 8 | df-co 5694 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
| 9 | df-co 5694 | . 2 ⊢ (𝐵 ∘ 𝐷) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)} | |
| 10 | 7, 8, 9 | 3sstr4g 4037 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) |
| 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: trrelssd 15012 ustund 24230 bj-imdirco 37191 relexpss1d 43718 |
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