Theorem bj-xtagex 37357

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 3454	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
2		bj-tagex 37355	. . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
3	1, 2	sylib 220	. 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V)
4		bj-xpexg2 37328	. 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 2121  Vcvv 3433   × cxp 5619  tag bj-ctag 37342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-opab 5138  df-xp 5627  df-rel 5628  df-bj-sngl 37334  df-bj-tag 37343

This theorem is referenced by:  bj-1uplex  37376  bj-2uplex  37390