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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 4626 | . . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦)) |
3 | coss12d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssbrd 4626 | . . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) |
5 | 2, 4 | anim12d 584 | . . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
6 | 5 | eximdv 1833 | . . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
7 | 6 | ssopab2dv 4929 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}) |
8 | df-co 5047 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
9 | df-co 5047 | . 2 ⊢ (𝐵 ∘ 𝐷) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)} | |
10 | 7, 8, 9 | 3sstr4g 3609 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ⊆ wss 3540 class class class wbr 4583 {copab 4642 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: trrelssd 13560 relexpss1d 37016 |
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