Theorem elimasn 4712

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.

Step	Hyp	Ref	Expression
1		elimasn.2	. . 3 ⊢ 𝐶 ∈ V
2		breq2 3789	. . 3 ⊢ (𝑥 = 𝐶 → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶))
3		elimasn.1	. . . 4 ⊢ 𝐵 ∈ V
4		imasng 4710	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
5	3, 4	ax-mp 7	. . 3 ⊢ (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}
6	1, 2, 5	elab2 2741	. 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7		df-br 3786	. 2 ⊢ (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
8	6, 7	bitri 182	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

Syntax hints:   ↔ wb 103   = wceq 1284   ∈ wcel 1433  {cab 2067  Vcvv 2601  {csn 3398  ⟨cop 3401   class class class wbr 3785   “ cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  elimasng  4713  dfco2  4840  dfco2a  4841  ressn  4878