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Theorem elsnres 4590
Description: Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
Hypothesis
Ref Expression
elsnres.1 𝐶 V
Assertion
Ref Expression
elsnres (A (B ↾ {𝐶}) ↔ y(A = ⟨𝐶, y𝐶, y B))
Distinct variable groups:   y,A   y,B   y,𝐶

Proof of Theorem elsnres
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elres 4589 . 2 (A (B ↾ {𝐶}) ↔ x {𝐶}y(A = ⟨x, yx, y B))
2 rexcom4 2571 . 2 (x {𝐶}y(A = ⟨x, yx, y B) ↔ yx {𝐶} (A = ⟨x, yx, y B))
3 elsnres.1 . . . 4 𝐶 V
4 opeq1 3540 . . . . . 6 (x = 𝐶 → ⟨x, y⟩ = ⟨𝐶, y⟩)
54eqeq2d 2048 . . . . 5 (x = 𝐶 → (A = ⟨x, y⟩ ↔ A = ⟨𝐶, y⟩))
64eleq1d 2103 . . . . 5 (x = 𝐶 → (⟨x, y B ↔ ⟨𝐶, y B))
75, 6anbi12d 442 . . . 4 (x = 𝐶 → ((A = ⟨x, yx, y B) ↔ (A = ⟨𝐶, y𝐶, y B)))
83, 7rexsn 3406 . . 3 (x {𝐶} (A = ⟨x, yx, y B) ↔ (A = ⟨𝐶, y𝐶, y B))
98exbii 1493 . 2 (yx {𝐶} (A = ⟨x, yx, y B) ↔ y(A = ⟨𝐶, y𝐶, y B))
101, 2, 93bitri 195 1 (A (B ↾ {𝐶}) ↔ y(A = ⟨𝐶, y𝐶, y B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551  {csn 3367  cop 3370  cres 4290
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-sbc 2759  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: (None)
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