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Theorem relssres 4571
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 2534 . . . . . . . . 9 x V
3 vex 2534 . . . . . . . . 9 y V
42, 3opeldm 4461 . . . . . . . 8 (⟨x, y Ax dom A)
5 ssel 2912 . . . . . . . 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 4540 . . . . . 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 4355 . . 3 ((Rel A dom AB) → A ⊆ (AB))
12 resss 4558 . . 3 (AB) ⊆ A
1311, 12jctil 295 . 2 ((Rel A dom AB) → ((AB) ⊆ A A ⊆ (AB)))
14 eqss 2933 . 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 1226   wcel 1370  wss 2890  cop 3349  dom cdm 4268  cres 4270  Rel wrel 4273
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-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-dm 4278  df-res 4280
This theorem is referenced by:  resdm  4572  resid  4585  fnresdm  4930  f1ompt  5241
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