Theorem bj-seex 36901

Description: Version of seex 5642 with a disjoint variable condition replaced by a nonfreeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)

Hypothesis

Ref	Expression
bj-seex.nf	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
bj-seex	⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem bj-seex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 5636	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V)
2		bj-seex.nf	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3	2	nfeq2 2922	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵
4		breq2 5145	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
5	3, 4	rabbid 3463	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})
6	5	eleq1d 2825	. . 3 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V))
7	6	rspccva 3620	. 2 ⊢ ((∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)
8	1, 7	sylanb 581	1 ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2889  ∀wral 3060  {crab 3435  Vcvv 3479   class class class wbr 5141   Se wse 5633

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-se 5636

This theorem is referenced by: (None)