Theorem axrep6g 5248

Description: axrep6 5246 in class notation. It is equivalent to both ax-rep 5237 and abrexexg 7942, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)

Assertion

Ref	Expression
axrep6g	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem axrep6g

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 3297	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
2	1	abbidv 2796	. . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓})
3	2	eleq1d 2814	. . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))
4	3	imbi2d 340	. . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)))
5		axrep6 5246	. . . 4 ⊢ (∀𝑥∃*𝑦𝜓 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓))
6		abbi 2795	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ 𝑦 ∈ 𝑤} = {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓})
7		abid2 2866	. . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} = 𝑤
8		vex 3454	. . . . . . 7 ⊢ 𝑤 ∈ V
9	7, 8	eqeltri 2825	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} ∈ V
10	6, 9	eqeltrrdi 2838	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
11	10	exlimiv 1930	. . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
12	5, 11	syl 17	. . 3 ⊢ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
13	4, 12	vtoclg 3523	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))
14	13	imp 406	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  {cab 2708  ∃wrex 3054  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-rep 5237

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3452

This theorem is referenced by:  funimaexg  6606  abrexexg  7942  permaxrep  45003