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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 5100 | . . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦)) |
3 | coss12d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssbrd 5100 | . . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) |
5 | 2, 4 | anim12d 608 | . . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
6 | 5 | eximdv 1909 | . . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
7 | 6 | ssopab2dv 5429 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}) |
8 | df-co 5557 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
9 | df-co 5557 | . 2 ⊢ (𝐵 ∘ 𝐷) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)} | |
10 | 7, 8, 9 | 3sstr4g 4009 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 ⊆ wss 3933 class class class wbr 5057 {copab 5119 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-co 5557 |
This theorem is referenced by: trrelssd 14321 ustund 22757 relexpss1d 39928 |
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