Theorem uniixp 6777

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 6776	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
2		fssxp 5422	. . . . 5 ⊢ (𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
3	1, 2	syl 14	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
4		velpw 3609	. . . 4 ⊢ (𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
5	3, 4	sylibr 134	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ssriv 3184	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)
7		sspwuni 3998	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵))
8	6, 7	mpbi 145	1 ⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2164   ⊆ wss 3154  𝒫 cpw 3602  ∪ cuni 3836  ∪ ciun 3913   × cxp 4658  ⟶wf 5251  Xcixp 6754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ixp 6755

This theorem is referenced by:  ixpexgg  6778