Theorem bj-seex 32111

Description: Version of seex 5001 with a dv condition replaced by a non-freeness 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 4998	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V)
2		bj-seex.nf	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3	2	nfeq2 2766	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵
4		breq2 4587	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
5	3, 4	bj-rabbid 32107	. . . 4 ⊢ (𝑦 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})
6	5	eleq1d 2672	. . 3 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V))
7	6	rspccva 3281	. 2 ⊢ ((∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)
8	1, 7	sylanb 488	1 ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   Se wse 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-se 4998

This theorem is referenced by: (None)