Theorem elrelimasn 6090

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 6089	. . 3 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
2	1	eleq2d 2851	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
3		brrelex2 5717	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
4	3	ex 418	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
5		breq2 5115	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
6	5	elab3g 3646	. . 3 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
7	4, 6	syl 18	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
8	2, 7	bitrd 282	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  {cab 2743  Vcvv 3457  {csn 4591   class class class wbr 5111   “ cima 5666  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  eliniseg2  6110  dprd2dlem2  20158  dprd2dlem1  20159  dprd2da  20160  dprd2d2  20162  dpjfval  20173  ustuqtop4  24454  utop2nei  24460  utop3cls  24461  ucncn  24494  extdgval  34109  cnambfre  38378  frege133d  44551  nzin  45088