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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 5073 | . . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦)) |
3 | coss12d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssbrd 5073 | . . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧)) |
5 | 2, 4 | anim12d 611 | . . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
6 | 5 | eximdv 1918 | . . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧))) |
7 | 6 | ssopab2dv 5403 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}) |
8 | df-co 5528 | . 2 ⊢ (𝐴 ∘ 𝐶) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} | |
9 | df-co 5528 | . 2 ⊢ (𝐵 ∘ 𝐷) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)} | |
10 | 7, 8, 9 | 3sstr4g 3960 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1781 ⊆ wss 3881 class class class wbr 5030 {copab 5092 ∘ ccom 5523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 df-co 5528 |
This theorem is referenced by: trrelssd 14324 ustund 22827 bj-imdirco 34605 relexpss1d 40406 |
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