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Theorem elres 4573
Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
Assertion
Ref Expression
elres (A (B𝐶) ↔ x 𝐶 y(A = ⟨x, yx, y B))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem elres
StepHypRef Expression
1 relres 4566 . . . . 5 Rel (B𝐶)
2 elrel 4369 . . . . 5 ((Rel (B𝐶) A (B𝐶)) → xy A = ⟨x, y⟩)
31, 2mpan 402 . . . 4 (A (B𝐶) → xy A = ⟨x, y⟩)
4 eleq1 2082 . . . . . . . . 9 (A = ⟨x, y⟩ → (A (B𝐶) ↔ ⟨x, y (B𝐶)))
54biimpd 132 . . . . . . . 8 (A = ⟨x, y⟩ → (A (B𝐶) → ⟨x, y (B𝐶)))
6 vex 2538 . . . . . . . . . . 11 y V
76opelres 4544 . . . . . . . . . 10 (⟨x, y (B𝐶) ↔ (⟨x, y B x 𝐶))
87biimpi 113 . . . . . . . . 9 (⟨x, y (B𝐶) → (⟨x, y B x 𝐶))
98ancomd 254 . . . . . . . 8 (⟨x, y (B𝐶) → (x 𝐶 x, y B))
105, 9syl6com 31 . . . . . . 7 (A (B𝐶) → (A = ⟨x, y⟩ → (x 𝐶 x, y B)))
1110ancld 308 . . . . . 6 (A (B𝐶) → (A = ⟨x, y⟩ → (A = ⟨x, y (x 𝐶 x, y B))))
12 an12 483 . . . . . 6 ((A = ⟨x, y (x 𝐶 x, y B)) ↔ (x 𝐶 (A = ⟨x, yx, y B)))
1311, 12syl6ib 150 . . . . 5 (A (B𝐶) → (A = ⟨x, y⟩ → (x 𝐶 (A = ⟨x, yx, y B))))
14132eximdv 1744 . . . 4 (A (B𝐶) → (xy A = ⟨x, y⟩ → xy(x 𝐶 (A = ⟨x, yx, y B))))
153, 14mpd 13 . . 3 (A (B𝐶) → xy(x 𝐶 (A = ⟨x, yx, y B)))
16 rexcom4 2554 . . . 4 (x 𝐶 y(A = ⟨x, yx, y B) ↔ yx 𝐶 (A = ⟨x, yx, y B))
17 df-rex 2290 . . . . 5 (x 𝐶 (A = ⟨x, yx, y B) ↔ x(x 𝐶 (A = ⟨x, yx, y B)))
1817exbii 1478 . . . 4 (yx 𝐶 (A = ⟨x, yx, y B) ↔ yx(x 𝐶 (A = ⟨x, yx, y B)))
19 excom 1536 . . . 4 (yx(x 𝐶 (A = ⟨x, yx, y B)) ↔ xy(x 𝐶 (A = ⟨x, yx, y B)))
2016, 18, 193bitri 195 . . 3 (x 𝐶 y(A = ⟨x, yx, y B) ↔ xy(x 𝐶 (A = ⟨x, yx, y B)))
2115, 20sylibr 137 . 2 (A (B𝐶) → x 𝐶 y(A = ⟨x, yx, y B))
227simplbi2com 1312 . . . . . 6 (x 𝐶 → (⟨x, y B → ⟨x, y (B𝐶)))
234biimprd 147 . . . . . 6 (A = ⟨x, y⟩ → (⟨x, y (B𝐶) → A (B𝐶)))
2422, 23syl9 66 . . . . 5 (x 𝐶 → (A = ⟨x, y⟩ → (⟨x, y BA (B𝐶))))
2524impd 242 . . . 4 (x 𝐶 → ((A = ⟨x, yx, y B) → A (B𝐶)))
2625exlimdv 1682 . . 3 (x 𝐶 → (y(A = ⟨x, yx, y B) → A (B𝐶)))
2726rexlimiv 2405 . 2 (x 𝐶 y(A = ⟨x, yx, y B) → A (B𝐶))
2821, 27impbii 117 1 (A (B𝐶) ↔ x 𝐶 y(A = ⟨x, yx, y B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353  cres 4274  Rel wrel 4277
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793  df-xp 4278  df-rel 4279  df-res 4284
This theorem is referenced by:  elsnres  4574
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