Theorem bj-xtagex 37012

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 3485	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
2		bj-tagex 37010	. . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
3	1, 2	sylib 218	. 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V)
4		bj-xpexg2 36983	. 2 ⊢ (𝐴 ∈ 𝑉 → (tag 𝐵 ∈ V → (𝐴 × tag 𝐵) ∈ V))
5	3, 4	syl5 34	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → (𝐴 × tag 𝐵) ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464   × cxp 5657  tag bj-ctag 36997

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-opab 5187  df-xp 5665  df-rel 5666  df-bj-sngl 36989  df-bj-tag 36998

This theorem is referenced by:  bj-1uplex  37031  bj-2uplex  37045