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Theorem coss1 4434
 Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
coss1 (AB → (A𝐶) ⊆ (B𝐶))

Proof of Theorem coss1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6 (ABAB)
21ssbrd 3796 . . . . 5 (AB → (yAzyBz))
32anim2d 320 . . . 4 (AB → ((x𝐶y yAz) → (x𝐶y yBz)))
43eximdv 1757 . . 3 (AB → (y(x𝐶y yAz) → y(x𝐶y yBz)))
54ssopab2dv 4006 . 2 (AB → {⟨x, z⟩ ∣ y(x𝐶y yAz)} ⊆ {⟨x, z⟩ ∣ y(x𝐶y yBz)})
6 df-co 4297 . 2 (A𝐶) = {⟨x, z⟩ ∣ y(x𝐶y yAz)}
7 df-co 4297 . 2 (B𝐶) = {⟨x, z⟩ ∣ y(x𝐶y yBz)}
85, 6, 73sstr4g 2980 1 (AB → (A𝐶) ⊆ (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ⊆ wss 2911   class class class wbr 3755  {copab 3808   ∘ ccom 4292 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-co 4297 This theorem is referenced by:  coeq1  4436  funss  4863  tposss  5802
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