Theorem bdinex1g 16841

Description: Bounded version of inex1g 4264. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bdinex1g.bd	⊢ BOUNDED 𝐵

Assertion

Ref	Expression
bdinex1g	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Proof of Theorem bdinex1g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3425	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2307	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		bdinex1g.bd	. . 3 ⊢ BOUNDED 𝐵
4		vex 2824	. . 3 ⊢ 𝑥 ∈ V
5	3, 4	bdinex1 16839	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
6	2, 5	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219  BOUNDED wbdc 16780

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bdsep 16824

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-bdc 16781

This theorem is referenced by: (None)