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Theorem elimasn 4990
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 4004 . . 3 (𝑥 = 𝐶 → (𝐵𝐴𝑥𝐵𝐴𝐶))
3 elimasn.1 . . . 4 𝐵 ∈ V
4 imasng 4988 . . . 4 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
53, 4ax-mp 5 . . 3 (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥}
61, 2, 5elab2 2885 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7 df-br 4001 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
86, 7bitri 184 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  {cab 2163  Vcvv 2737  {csn 3591  cop 3594   class class class wbr 4000  cima 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-cnv 4630  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635
This theorem is referenced by:  elimasng  4991  dfco2  5123  dfco2a  5124  ressn  5164
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