Theorem elrelimasn 6106

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 6105	. . 3 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
2	1	eleq2d 2825	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
3		brrelex2 5743	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
4	3	ex 412	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
5		breq2 5152	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
6	5	elab3g 3688	. . 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 2106  {cab 2712  Vcvv 3478  {csn 4631   class class class wbr 5148   “ cima 5692  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702

This theorem is referenced by:  eliniseg2  6127  dprd2dlem2  20075  dprd2dlem1  20076  dprd2da  20077  dprd2d2  20079  dpjfval  20090  ustuqtop4  24269  utop2nei  24275  utop3cls  24276  ucncn  24310  extdgval  33682  cnambfre  37655  frege133d  43755  nzin  44314