xpkexg - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > xpkexg		GIF version

Theorem xpkexg 4289

Description: The Kuratowski cross product of two sets is a set. (Contributed by SF, 13-Jan-2015.)

**Assertion**
Ref	Expression
xpkexg	⊢ ((A ∈ V ∧ B ∈ W) → (A ×_k B) ∈ V)

**Proof of Theorem xpkexg**
Step	Hyp	Ref	Expression
1		cnvkxpk 4277	. . 3 ⊢ ^◡_k(V ×_k A) = (A ×_k V)
2		xpkvexg 4286	. . . 4 ⊢ (A ∈ V → (V ×_k A) ∈ V)
3		cnvkexg 4287	. . . 4 ⊢ ((V ×_k A) ∈ V → ^◡_k(V ×_k A) ∈ V)
4	2, 3	syl 15	. . 3 ⊢ (A ∈ V → ^◡_k(V ×_k A) ∈ V)
5	1, 4	syl5eqelr 2438	. 2 ⊢ (A ∈ V → (A ×_k V) ∈ V)
6		xpkvexg 4286	. 2 ⊢ (B ∈ W → (V ×_k B) ∈ V)
7		inxpk 4278	. . . 4 ⊢ ((A ×_k V) ∩ (V ×_k B)) = ((A ∩ V) ×_k (V ∩ B))
8		inv1 3578	. . . . 5 ⊢ (A ∩ V) = A
9		incom 3449	. . . . . 6 ⊢ (V ∩ B) = (B ∩ V)
10		inv1 3578	. . . . . 6 ⊢ (B ∩ V) = B
11	9, 10	eqtri 2373	. . . . 5 ⊢ (V ∩ B) = B
12	8, 11	xpkeq12i 4204	. . . 4 ⊢ ((A ∩ V) ×_k (V ∩ B)) = (A ×_k B)
13	7, 12	eqtri 2373	. . 3 ⊢ ((A ×_k V) ∩ (V ×_k B)) = (A ×_k B)
14		inexg 4101	. . 3 ⊢ (((A ×_k V) ∈ V ∧ (V ×_k B) ∈ V) → ((A ×_k V) ∩ (V ×_k B)) ∈ V)
15	13, 14	syl5eqelr 2438	. 2 ⊢ (((A ×_k V) ∈ V ∧ (V ×_k B) ∈ V) → (A ×_k B) ∈ V)
16	5, 6, 15	syl2an 463	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ×_k B) ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 Vcvv 2860 ∩ cin 3209 ×_k cxpk 4175 ^◡_kccnvk 4176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-sn 4088

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-opk 4059 df-xpk 4186 df-cnvk 4187

This theorem is referenced by: xpkex 4290 uni1exg 4293 imakexg 4300 pw1exg 4303 pwexg 4329