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Mirrors > Home > NFE Home > Th. List > coss2 | GIF version |
Description: Subclass theorem for composition. (Contributed by set.mm contributors, 5-Apr-2013.) |
Ref | Expression |
---|---|
coss2 | ⊢ (A ⊆ B → (C ∘ A) ⊆ (C ∘ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . 6 ⊢ (A ⊆ B → A ⊆ B) | |
2 | 1 | ssbrd 4680 | . . . . 5 ⊢ (A ⊆ B → (xAy → xBy)) |
3 | 2 | anim1d 547 | . . . 4 ⊢ (A ⊆ B → ((xAy ∧ yCz) → (xBy ∧ yCz))) |
4 | 3 | eximdv 1622 | . . 3 ⊢ (A ⊆ B → (∃y(xAy ∧ yCz) → ∃y(xBy ∧ yCz))) |
5 | 4 | ssopab2dv 4715 | . 2 ⊢ (A ⊆ B → {〈x, z〉 ∣ ∃y(xAy ∧ yCz)} ⊆ {〈x, z〉 ∣ ∃y(xBy ∧ yCz)}) |
6 | df-co 4726 | . 2 ⊢ (C ∘ A) = {〈x, z〉 ∣ ∃y(xAy ∧ yCz)} | |
7 | df-co 4726 | . 2 ⊢ (C ∘ B) = {〈x, z〉 ∣ ∃y(xBy ∧ yCz)} | |
8 | 5, 6, 7 | 3sstr4g 3312 | 1 ⊢ (A ⊆ B → (C ∘ A) ⊆ (C ∘ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 ⊆ wss 3257 {copab 4622 class class class wbr 4639 ∘ ccom 4721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-co 4726 |
This theorem is referenced by: coeq2 4875 funss 5126 |
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