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Theorem eliniseg 4618
Description: Membership in an initial segment. The idiom (A “ {B}), meaning {xxAB}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
eliniseg.1 𝐶 V
Assertion
Ref Expression
eliniseg (B 𝑉 → (𝐶 (A “ {B}) ↔ 𝐶AB))

Proof of Theorem eliniseg
StepHypRef Expression
1 eliniseg.1 . 2 𝐶 V
2 elimasng 4616 . . . 4 ((B 𝑉 𝐶 V) → (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A))
3 df-br 3735 . . . 4 (BA𝐶 ↔ ⟨B, 𝐶 A)
42, 3syl6bbr 187 . . 3 ((B 𝑉 𝐶 V) → (𝐶 (A “ {B}) ↔ BA𝐶))
5 brcnvg 4439 . . 3 ((B 𝑉 𝐶 V) → (BA𝐶𝐶AB))
64, 5bitrd 177 . 2 ((B 𝑉 𝐶 V) → (𝐶 (A “ {B}) ↔ 𝐶AB))
71, 6mpan2 403 1 (B 𝑉 → (𝐶 (A “ {B}) ↔ 𝐶AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1370  Vcvv 2531  {csn 3346  cop 3349   class class class wbr 3734  ccnv 4267  cima 4271
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-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  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-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281
This theorem is referenced by:  epini  4619  iniseg  4620  dfco2a  4744  isoini  5378
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