Theorem elabrex 7190

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 1546	. . . 4 ⊢ ⊤
2		csbeq1a 3864	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
3	2	equcoms 2022	. . . . . 6 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
4		trud 1552	. . . . . 6 ⊢ (𝑧 = 𝑥 → ⊤)
5	3, 4	2thd 265	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤))
6	5	rspcev 3577	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
7	1, 6	mpan2 692	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
8		elabrex.1	. . . 4 ⊢ 𝐵 ∈ V
9		eqeq1 2741	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10	9	rexbidv 3161	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
11	8, 10	elab 3635	. . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
12	7, 11	sylibr 234	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵})
13		nfv 1916	. . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵
14		nfcsb1v 3874	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
16	2	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
17	13, 15, 16	cbvrexw 3280	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
18	17	abbii 2804	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}
19	12, 18	eleqtrrdi 2848	1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  {cab 2715  ∃wrex 3061  Vcvv 3441  ⦋csb 3850

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-sbc 3742  df-csb 3851

This theorem is referenced by:  eusvobj2  7352  lss1d  20918  prdsxmetlem  24316  prdsbl  24439  itg2monolem1  25711  heibor1  37982  dihglblem5  41595