Theorem uniixp 8667

Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
uniixp	⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem uniixp

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpf 8666	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
2		fssxp 6612	. . . . 5 ⊢ (𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
4		velpw 4535	. . . 4 ⊢ (𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
5	3, 4	sylibr 233	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ssriv 3921	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)
7		sspwuni 5025	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
8	6, 7	mpbi 229	1 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2108   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ∪ ciun 4921   × cxp 5578  ⟶wf 6414  Xcixp 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ixp 8644

This theorem is referenced by:  ixpexg  8668