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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 5193 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) | |
2 | 1 | anim1d 609 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
3 | 2 | eximdv 1918 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧))) |
4 | 3 | ssopab2dv 5552 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}) |
5 | df-co 5686 | . 2 ⊢ (𝐶 ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} | |
6 | df-co 5686 | . 2 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)} | |
7 | 4, 5, 6 | 3sstr4g 4028 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∃wex 1779 ⊆ wss 3949 class class class wbr 5149 {copab 5211 ∘ ccom 5681 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3956 df-ss 3966 df-br 5150 df-opab 5212 df-co 5686 |
This theorem is referenced by: coeq2 5859 funss 6568 tposss 8216 dftpos4 8234 ttrclco 9717 frmin 9748 frrlem16 9757 rtrclreclem4 15014 tsrdir 18563 mvdco 19356 ustex2sym 23943 ustex3sym 23944 ustneism 23950 trust 23956 utop2nei 23977 neipcfilu 24023 fcoinver 32100 trclubgNEW 42673 trrelsuperrel2dg 42726 |
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