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Theorem resco 4768
Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resco ((AB) ↾ 𝐶) = (A ∘ (B𝐶))

Proof of Theorem resco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4582 . 2 Rel ((AB) ↾ 𝐶)
2 relco 4762 . 2 Rel (A ∘ (B𝐶))
3 vex 2554 . . . . . 6 x V
4 vex 2554 . . . . . 6 y V
53, 4brco 4449 . . . . 5 (x(AB)yz(xBz zAy))
65anbi1i 431 . . . 4 ((x(AB)y x 𝐶) ↔ (z(xBz zAy) x 𝐶))
7 19.41v 1779 . . . 4 (z((xBz zAy) x 𝐶) ↔ (z(xBz zAy) x 𝐶))
8 an32 496 . . . . . 6 (((xBz zAy) x 𝐶) ↔ ((xBz x 𝐶) zAy))
9 vex 2554 . . . . . . . 8 z V
109brres 4561 . . . . . . 7 (x(B𝐶)z ↔ (xBz x 𝐶))
1110anbi1i 431 . . . . . 6 ((x(B𝐶)z zAy) ↔ ((xBz x 𝐶) zAy))
128, 11bitr4i 176 . . . . 5 (((xBz zAy) x 𝐶) ↔ (x(B𝐶)z zAy))
1312exbii 1493 . . . 4 (z((xBz zAy) x 𝐶) ↔ z(x(B𝐶)z zAy))
146, 7, 133bitr2i 197 . . 3 ((x(AB)y x 𝐶) ↔ z(x(B𝐶)z zAy))
154brres 4561 . . 3 (x((AB) ↾ 𝐶)y ↔ (x(AB)y x 𝐶))
163, 4brco 4449 . . 3 (x(A ∘ (B𝐶))yz(x(B𝐶)z zAy))
1714, 15, 163bitr4i 201 . 2 (x((AB) ↾ 𝐶)yx(A ∘ (B𝐶))y)
181, 2, 17eqbrriv 4378 1 ((AB) ↾ 𝐶) = (A ∘ (B𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390   class class class wbr 3755  cres 4290  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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297  df-res 4300
This theorem is referenced by:  cocnvcnv2  4775  coires1  4781  relcoi1  4792  dftpos2  5817
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