Theorem elrelimasn 5133

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 variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 5110	. . . . . 6 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵))
2	1	ibi 176	. . . . 5 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵)
3		rexm 3613	. . . . 5 ⊢ (∃𝑦 ∈ {𝐴}𝑦𝑅𝐵 → ∃𝑦 𝑦 ∈ {𝐴})
4		elsni 3712	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
5	4	eximi 1649	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝐴} → ∃𝑦 𝑦 = 𝐴)
6	2, 3, 5	3syl 17	. . . 4 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 𝑦 = 𝐴)
7		isset 2822	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
8	6, 7	sylibr 134	. . 3 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V)
9	8	a1i 9	. 2 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) → 𝐴 ∈ V))
10		brrelex1 4794	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
11	10	ex 115	. 2 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V))
12		imasng 5132	. . . . 5 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
13	12	eleq2d 2304	. . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
14		brrelex2 4796	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
15	14	ex 115	. . . . 5 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
16		breq2 4118	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
17	16	elab3g 2971	. . . . 5 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
18	15, 17	syl 14	. . . 4 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
19	13, 18	sylan9bbr 463	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
20	19	ex 115	. 2 ⊢ (Rel 𝑅 → (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)))
21	9, 11, 20	pm5.21ndd 713	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∃wrex 2523  Vcvv 2815  {csn 3694   class class class wbr 4114   “ cima 4757  Rel wrel 4759

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  eliniseg2  5147