Theorem bj-seex 35590

Description: Version of seex 5628 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 5622	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V)
2		bj-seex.nf	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3	2	nfeq2 2919	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵
4		breq2 5142	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
5	3, 4	rabbid 3456	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})
6	5	eleq1d 2817	. . 3 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V))
7	6	rspccva 3605	. 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 396   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2882  ∀wral 3060  {crab 3429  Vcvv 3470   class class class wbr 5138   Se wse 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-br 5139  df-se 5622

This theorem is referenced by: (None)