Theorem elrelimasn 5148

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 5125	. . . . . 6 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵))
2	1	ibi 176	. . . . 5 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵)
3		rexm 3624	. . . . 5 ⊢ (∃𝑦 ∈ {𝐴}𝑦𝑅𝐵 → ∃𝑦 𝑦 ∈ {𝐴})
4		elsni 3723	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
5	4	eximi 1653	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝐴} → ∃𝑦 𝑦 = 𝐴)
6	2, 3, 5	3syl 17	. . . 4 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 𝑦 = 𝐴)
7		isset 2828	. . . 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 4809	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
11	10	ex 115	. 2 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V))
12		imasng 5147	. . . . 5 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
13	12	eleq2d 2308	. . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
14		brrelex2 4811	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
15	14	ex 115	. . . . 5 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
16		breq2 4129	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
17	16	elab3g 2977	. . . . 5 ⊢ ((𝐴𝑅𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
18	15, 17	syl 14	. . . 4 ⊢ (Rel 𝑅 → (𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
19	13, 18	sylan9bbr 467	. . 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 717	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821  {csn 3705   class class class wbr 4125   “ cima 4772  Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  eliniseg2  5162