Theorem elrelimasn 6046

Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)

Assertion

Ref	Expression
elrelimasn	⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elrelimasn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relimasn 6045	. . 3 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
2	1	eleq2d 2814	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
3		brrelex2 5685	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
4	3	ex 412	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
5		breq2 5106	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
6	5	elab3g 3649	. . 3 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
7	4, 6	syl 17	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
8	2, 7	bitrd 279	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  {cab 2707  Vcvv 3444  {csn 4585   class class class wbr 5102   “ cima 5634  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  eliniseg2  6066  dprd2dlem2  19956  dprd2dlem1  19957  dprd2da  19958  dprd2d2  19960  dpjfval  19971  ustuqtop4  24165  utop2nei  24171  utop3cls  24172  ucncn  24205  extdgval  33642  cnambfre  37655  frege133d  43747  nzin  44300