Theorem bj-xtagex 32170

Description: The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)

Assertion

Ref	Expression
bj-xtagex	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V))

Proof of Theorem bj-xtagex

Step	Hyp	Ref	Expression
1		elex 3185	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
2		bj-tagex 32168	. . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
3	1, 2	sylib 207	. 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V)
4		bj-xpexg2 32140	. 2 ⊢ (𝐴 ∈ 𝑉 → (tag 𝐵 ∈ V → (𝐴 × tag 𝐵) ∈ V))
5	3, 4	syl5 33	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V))

Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173   × cxp 5036  tag bj-ctag 32155

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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-opab 4644  df-xp 5044  df-rel 5045  df-bj-sngl 32147  df-bj-tag 32156

This theorem is referenced by:  bj-1uplex  32189  bj-2uplex  32203