Theorem djussxp 5825

Description: Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Assertion

Ref	Expression
djussxp	⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem djussxp

Step	Hyp	Ref	Expression
1		iunss 5003	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2		snssi 4746	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
3		ssv 3955	. . 3 ⊢ 𝐵 ⊆ V
4		xpss12 5670	. . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
5	2, 3, 4	sylancl 598	. 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
6	1, 5	mprgbir 3083	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3899  {csn 4584  ∪ ciun 4951   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-sn 4585  df-iun 4953  df-opab 5168  df-xp 5661

This theorem is used by:  djudisj  6159  iundom2g  10549