Theorem axrep6g 5245

Description: axrep6 5241 in class notation. It is equivalent to both ax-rep 5232 and abrexexg 7946, 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 3319	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
2	1	abbidv 2831	. . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓})
3	2	eleq1d 2850	. . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))
4	3	imbi2d 343	. . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)))
5		axrep6 5241	. . . 4 ⊢ (∀𝑥∃*𝑦𝜓 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓))
6		abbi 2830	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ 𝑦 ∈ 𝑤} = {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓})
7		abid2 2902	. . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} = 𝑤
8		vex 3461	. . . . . . 7 ⊢ 𝑤 ∈ V
9	7, 8	eqeltri 2861	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} ∈ V
10	6, 9	eqeltrrdi 2874	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
11	10	exlimiv 1953	. . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
12	5, 11	syl 18	. . 3 ⊢ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V)
13	4, 12	vtoclg 3525	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))
14	13	imp 411	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃*wmo 2567  {cab 2743  ∃wrex 3089  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-rep 5232

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3090  df-v 3459

This theorem is referenced by:  funimaexg  6612  abrexexg  7946  permaxrep  45580