Theorem elabrexg 7221

Description: Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

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

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

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabrexg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1563	. . . . 5 ⊢ ⊤
2		csbeq1a 3866	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
3	2	equcoms 2039	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
4		trud 1569	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ⊤)
5	3, 4	2thd 267	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤))
6	5	rspcev 3581	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
7	1, 6	mpan2 701	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
8	7	adantr 484	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
9		eqeq1 2765	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
10	9	rexbidv 3185	. . . . 5 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
11	10	elabg 3635	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
12	11	adantl 485	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵))
13	8, 12	mpbird 259	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵})
14		nfv 1933	. . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵
15		nfcsb1v 3876	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
16	15	nfeq2 2940	. . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵
17	2	eqeq2d 2772	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵))
18	14, 16, 17	cbvrexw 3304	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)
19	18	abbii 2828	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}
20	13, 19	eleqtrrdi 2872	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141  {cab 2739  ∃wrex 3085  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-sbc 3745  df-csb 3853

This theorem is referenced by:  upbdrech  45837  ssfiunibd  45841