Theorem elrelimasn 6075

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 6074	. . 3 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
2	1	eleq2d 2848	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
3		brrelex2 5701	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
4	3	ex 416	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
5		breq2 5104	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
6	5	elab3g 3644	. . 3 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
7	4, 6	syl 17	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
8	2, 7	bitrd 281	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2142  {cab 2740  Vcvv 3454  {csn 4582   class class class wbr 5100   “ cima 5650  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  eliniseg2  6095  dprd2dlem2  20082  dprd2dlem1  20083  dprd2da  20084  dprd2d2  20086  dpjfval  20097  ustuqtop4  24304  utop2nei  24310  utop3cls  24311  ucncn  24344  extdgval  33950  cnambfre  38167  frege133d  44341  nzin  44894