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Theorem relssres 4591
Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
relssres ((Rel A dom AB) → (AB) = A)

Proof of Theorem relssres
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . 4 ((Rel A dom AB) → Rel A)
2 vex 2554 . . . . . . . . 9 x V
3 vex 2554 . . . . . . . . 9 y V
42, 3opeldm 4481 . . . . . . . 8 (⟨x, y Ax dom A)
5 ssel 2933 . . . . . . . 8 (dom AB → (x dom Ax B))
64, 5syl5 28 . . . . . . 7 (dom AB → (⟨x, y Ax B))
76ancld 308 . . . . . 6 (dom AB → (⟨x, y A → (⟨x, y A x B)))
83opelres 4560 . . . . . 6 (⟨x, y (AB) ↔ (⟨x, y A x B))
97, 8syl6ibr 151 . . . . 5 (dom AB → (⟨x, y A → ⟨x, y (AB)))
109adantl 262 . . . 4 ((Rel A dom AB) → (⟨x, y A → ⟨x, y (AB)))
111, 10relssdv 4375 . . 3 ((Rel A dom AB) → A ⊆ (AB))
12 resss 4578 . . 3 (AB) ⊆ A
1311, 12jctil 295 . 2 ((Rel A dom AB) → ((AB) ⊆ A A ⊆ (AB)))
14 eqss 2954 . 2 ((AB) = A ↔ ((AB) ⊆ A A ⊆ (AB)))
1513, 14sylibr 137 1 ((Rel A dom AB) → (AB) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wss 2911  cop 3370  dom cdm 4288  cres 4290  Rel wrel 4293
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-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-dm 4298  df-res 4300
This theorem is referenced by:  resdm  4592  resid  4605  fnresdm  4951  f1ompt  5263
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