Theorem elimasn 5103

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 4092	. . 3 ⊢ (𝑥 = 𝐶 → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶))
3		elimasn.1	. . . 4 ⊢ 𝐵 ∈ V
4		imasng 5101	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
5	3, 4	ax-mp 5	. . 3 ⊢ (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}
6	1, 2, 5	elab2 2954	. 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7		df-br 4089	. 2 ⊢ (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
8	6, 7	bitri 184	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {cab 2217  Vcvv 2802  {csn 3669  ⟨cop 3672   class class class wbr 4088   “ cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  elimasng  5104  dfco2  5236  dfco2a  5237  ressn  5277