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Theorem elimasn 5525
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 4689 . . 3 (𝑥 = 𝐶 → (𝐵𝐴𝑥𝐵𝐴𝐶))
3 elimasn.1 . . . 4 𝐵 ∈ V
4 imasng 5522 . . . 4 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
53, 4ax-mp 5 . . 3 (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥}
61, 2, 5elab2 3386 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7 df-br 4686 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
86, 7bitri 264 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  {csn 4210  cop 4216   class class class wbr 4685  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  elimasng  5526  dfco2  5672  dfco2a  5673  ressn  5709  funfvima3  6535  frxp  7332  marypha1lem  8380  gsum2dlem1  18415  gsum2dlem2  18416  gsum2d  18417  gsum2d2  18419  ovoliunlem1  23316  iunsnima  29556  dfcnv2  29604  gsummpt2co  29908  gsummpt2d  29909  dmscut  32043  scutf  32044  funpartfun  32175  areaquad  38119  dffrege76  38550  frege97  38571  frege98  38572  frege109  38583  frege110  38584  frege131  38605  frege133  38607
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