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Theorem elimasn 4635
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
elimasn.1 B V
elimasn.2 𝐶 V
Assertion
Ref Expression
elimasn (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A)

Proof of Theorem elimasn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elimasn.2 . . 3 𝐶 V
2 breq2 3759 . . 3 (x = 𝐶 → (BAxBA𝐶))
3 elimasn.1 . . . 4 B V
4 imasng 4633 . . . 4 (B V → (A “ {B}) = {xBAx})
53, 4ax-mp 7 . . 3 (A “ {B}) = {xBAx}
61, 2, 5elab2 2684 . 2 (𝐶 (A “ {B}) ↔ BA𝐶)
7 df-br 3756 . 2 (BA𝐶 ↔ ⟨B, 𝐶 A)
86, 7bitri 173 1 (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551  {csn 3367  cop 3370   class class class wbr 3755  cima 4291
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-eu 1900  df-mo 1901  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-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  elimasng  4636  dfco2  4763  dfco2a  4764  ressn  4801
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