Theorem bj-xtagex 36463

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 3489	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
2		bj-tagex 36461	. . 3 ⊢ (𝐵 ∈ V ↔ tag 𝐵 ∈ V)
3	1, 2	sylib 217	. 2 ⊢ (𝐵 ∈ 𝑊 → tag 𝐵 ∈ V)
4		bj-xpexg2 36434	. 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 2099  Vcvv 3470   × cxp 5671  tag bj-ctag 36448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-opab 5206  df-xp 5679  df-rel 5680  df-bj-sngl 36440  df-bj-tag 36449

This theorem is referenced by:  bj-1uplex  36482  bj-2uplex  36496