Theorem abrexex2g 6212

Description: Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
abrexex2g	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2g

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1552	. . . 4 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
2		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		nfs1v 1968	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
4	2, 3	nfrexw 2546	. . . 4 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
5		sbequ12 1795	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
6	5	rexbidv 2508	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
7	1, 4, 6	cbvab 2330	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
8		df-clab 2193	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
9	8	rexbii 2514	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
10	9	abbii 2322	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑}
11	7, 10	eqtr4i 2230	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}}
12		df-iun 3931	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}}
13		iunexg 6211	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V)
14	12, 13	eqeltrrid 2294	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} ∈ V)
15	11, 14	eqeltrid 2293	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  [wsb 1786   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486  Vcvv 2773  ∪ ciun 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284

This theorem is referenced by:  frecabex  6491  plyval  15248