Theorem elrelimasn 5048

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 5026	. . . . . 6 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵))
2	1	ibi 176	. . . . 5 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 ∈ {𝐴}𝑦𝑅𝐵)
3		rexm 3560	. . . . 5 ⊢ (∃𝑦 ∈ {𝐴}𝑦𝑅𝐵 → ∃𝑦 𝑦 ∈ {𝐴})
4		elsni 3651	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
5	4	eximi 1623	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝐴} → ∃𝑦 𝑦 = 𝐴)
6	2, 3, 5	3syl 17	. . . 4 ⊢ (𝐵 ∈ (𝑅 “ {𝐴}) → ∃𝑦 𝑦 = 𝐴)
7		isset 2778	. . . 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 4714	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
11	10	ex 115	. 2 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V))
12		imasng 5047	. . . . 5 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑥 ∣ 𝐴𝑅𝑥})
13	12	eleq2d 2275	. . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥 ∣ 𝐴𝑅𝑥}))
14		brrelex2 4716	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
15	14	ex 115	. . . . 5 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐵 ∈ V))
16		breq2 4048	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
17	16	elab3g 2924	. . . . 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 707	1 ⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  {cab 2191  ∃wrex 2485  Vcvv 2772  {csn 3633   class class class wbr 4044   “ cima 4678  Rel wrel 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by:  eliniseg2  5062