Theorem elabrex 7195

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Hypothesis

Ref	Expression
elabrex.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
elabrex	⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Distinct variable groups:   𝑦,𝐵   𝑥,𝑦,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem elabrex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1545	. . . 4 ⊢ ⊤
2		csbeq1a 3872	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
3	2	equcoms 2023	. . . . . 6 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
4		trud 1551	. . . . . 6 ⊢ (𝑧 = 𝑥 → ⊤)
5	3, 4	2thd 264	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤))
6	5	rspcev 3582	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
7	1, 6	mpan2 689	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
8		elabrex.1	. . . 4 ⊢ 𝐵 ∈ V
9		eqeq1 2735	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10	9	rexbidv 3171	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
11	8, 10	elab 3633	. . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
12	7, 11	sylibr 233	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵})
13		nfv 1917	. . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵
14		nfcsb1v 3883	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2919	. . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	2	eqeq2d 2742	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
17	13, 15, 16	cbvrexw 3288	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
18	17	abbii 2801	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}
19	12, 18	eleqtrrdi 2843	1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106  {cab 2708  ∃wrex 3069  Vcvv 3446  ⦋csb 3858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-sbc 3743  df-csb 3859

This theorem is referenced by:  eusvobj2  7354  lss1d  20481  prdsxmetlem  23758  prdsbl  23884  itg2monolem1  25152  heibor1  36342  dihglblem5  39834