Theorem uniixp 6820

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 6819	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)
2		fssxp 5452	. . . . 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 3627	. . . 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 3201	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ 𝒫 (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵)
7		sspwuni 4017	. 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 2177   ⊆ wss 3170  𝒫 cpw 3620  ∪ cuni 3855  ∪ ciun 3932   × cxp 4680  ⟶wf 5275  Xcixp 6797

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-fv 5287  df-ixp 6798

This theorem is referenced by:  ixpexgg  6821