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Theorem resco 4748
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 4562 . 2 Rel ((AB) ↾ 𝐶)
2 relco 4742 . 2 Rel (A ∘ (B𝐶))
3 vex 2534 . . . . . 6 x V
4 vex 2534 . . . . . 6 y V
53, 4brco 4429 . . . . 5 (x(AB)yz(xBz zAy))
65anbi1i 434 . . . 4 ((x(AB)y x 𝐶) ↔ (z(xBz zAy) x 𝐶))
7 19.41v 1760 . . . 4 (z((xBz zAy) x 𝐶) ↔ (z(xBz zAy) x 𝐶))
8 an32 484 . . . . . 6 (((xBz zAy) x 𝐶) ↔ ((xBz x 𝐶) zAy))
9 vex 2534 . . . . . . . 8 z V
109brres 4541 . . . . . . 7 (x(B𝐶)z ↔ (xBz x 𝐶))
1110anbi1i 434 . . . . . 6 ((x(B𝐶)z zAy) ↔ ((xBz x 𝐶) zAy))
128, 11bitr4i 176 . . . . 5 (((xBz zAy) x 𝐶) ↔ (x(B𝐶)z zAy))
1312exbii 1474 . . . 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 4541 . . 3 (x((AB) ↾ 𝐶)y ↔ (x(AB)y x 𝐶))
163, 4brco 4429 . . 3 (x(A ∘ (B𝐶))yz(x(B𝐶)z zAy))
1714, 15, 163bitr4i 201 . 2 (x((AB) ↾ 𝐶)yx(A ∘ (B𝐶))y)
181, 2, 17eqbrriv 4358 1 ((AB) ↾ 𝐶) = (A ∘ (B𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226  wex 1358   wcel 1370   class class class wbr 3734  cres 4270  ccom 4272
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-co 4277  df-res 4280
This theorem is referenced by:  cocnvcnv2  4755  coires1  4761  relcoi1  4772  dftpos2  5794
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