Theorem uniixp 4363

Description: The union of an infinite Cartesian product is included in a cross product.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem uniixp

Step	Hyp	Ref	Expression
1		eluni 2510	. . . 4 ⊢ (y ∈ ∪Xx ∈ A B ↔ ∃f(y ∈ f ⋀ f ∈ Xx ∈ A B))
2		ixpf 4362	. . . . . 6 ⊢ (f ∈ Xx ∈ A B → f:A–→∪x ∈ A B)
3	2	anim2i 335	. . . . 5 ⊢ ((y ∈ f ⋀ f ∈ Xx ∈ A B) → (y ∈ f ⋀ f:A–→∪x ∈ A B))
4	3	19.22i 1042	. . . 4 ⊢ (∃f(y ∈ f ⋀ f ∈ Xx ∈ A B) → ∃f(y ∈ f ⋀ f:A–→∪x ∈ A B))
5	1, 4	sylbi 199	. . 3 ⊢ (y ∈ ∪Xx ∈ A B → ∃f(y ∈ f ⋀ f:A–→∪x ∈ A B))
6		fssxp 3643	. . . . . 6 ⊢ (f:A–→∪x ∈ A B → f ⊆ (A × ∪x ∈ A B))
7	6	sseld 2070	. . . . 5 ⊢ (f:A–→∪x ∈ A B → (y ∈ f → y ∈ (A × ∪x ∈ A B)))
8	7	impcom 351	. . . 4 ⊢ ((y ∈ f ⋀ f:A–→∪x ∈ A B) → y ∈ (A × ∪x ∈ A B))
9	8	19.23aiv 1297	. . 3 ⊢ (∃f(y ∈ f ⋀ f:A–→∪x ∈ A B) → y ∈ (A × ∪x ∈ A B))
10	5, 9	syl 10	. 2 ⊢ (y ∈ ∪Xx ∈ A B → y ∈ (A × ∪x ∈ A B))
11	10	ssriv 2072	1 ⊢ ∪Xx ∈ A B ⊆ (A × ∪x ∈ A B)

Syntax hints:   ⋀ wa 223   ∈ wcel 960  ∃wex 982   ⊆ wss 2050  ∪cuni 2507  ∪ciun 2570   × cxp 3174  –→wf 3184  Xcixp 4353

This theorem is referenced by:  ixpexg 4364

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-ixp 4354

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