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Theorem elres 4589
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 4582 . . . . 5 Rel (B𝐶)
2 elrel 4385 . . . . 5 ((Rel (B𝐶) A (B𝐶)) → xy A = ⟨x, y⟩)
31, 2mpan 400 . . . 4 (A (B𝐶) → xy A = ⟨x, y⟩)
4 eleq1 2097 . . . . . . . . 9 (A = ⟨x, y⟩ → (A (B𝐶) ↔ ⟨x, y (B𝐶)))
54biimpd 132 . . . . . . . 8 (A = ⟨x, y⟩ → (A (B𝐶) → ⟨x, y (B𝐶)))
6 vex 2554 . . . . . . . . . . 11 y V
76opelres 4560 . . . . . . . . . 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 495 . . . . . 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 1759 . . . 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 2571 . . . 4 (x 𝐶 y(A = ⟨x, yx, y B) ↔ yx 𝐶 (A = ⟨x, yx, y B))
17 df-rex 2306 . . . . 5 (x 𝐶 (A = ⟨x, yx, y B) ↔ x(x 𝐶 (A = ⟨x, yx, y B)))
1817exbii 1493 . . . 4 (yx 𝐶 (A = ⟨x, yx, y B) ↔ yx(x 𝐶 (A = ⟨x, yx, y B)))
19 excom 1551 . . . 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 1330 . . . . . 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 1697 . . 3 (x 𝐶 → (y(A = ⟨x, yx, y B) → A (B𝐶)))
2726rexlimiv 2421 . 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 1242  wex 1378   wcel 1390  wrex 2301  cop 3370  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-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  elsnres  4590
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