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Mirrors > Home > MPE Home > Th. List > coss2 | Structured version Visualization version GIF version |
Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.) |
Ref | Expression |
---|---|
coss2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbr 5074 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) | |
2 | 1 | anim1d 613 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
3 | 2 | eximdv 1918 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
4 | 3 | ssopab2dv 5403 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}) |
5 | df-co 5528 | . 2 ⊢ (𝐶 ∘ 𝐴) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} | |
6 | df-co 5528 | . 2 ⊢ (𝐶 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)} | |
7 | 4, 5, 6 | 3sstr4g 3960 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
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: coeq2 5693 funss 6343 tposss 7876 dftpos4 7894 rtrclreclem4 14412 tsrdir 17840 mvdco 18565 ustex2sym 22822 ustex3sym 22823 ustneism 22829 trust 22835 utop2nei 22856 neipcfilu 22902 fcoinver 30370 trclubgNEW 40318 trrelsuperrel2dg 40372 |
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