Theorem djussxp 4749

Description: Disjoint union is a subset of a cross 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 3907	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V))
2		snssi 3717	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
3		ssv 3164	. . 3 ⊢ 𝐵 ⊆ V
4		xpss12 4711	. . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
5	2, 3, 4	sylancl 410	. 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V))
6	1, 5	mprgbir 2524	1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)

Syntax hints:   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  {csn 3576  ∪ ciun 3866   × cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-sn 3582  df-iun 3868  df-opab 4044  df-xp 4610

This theorem is referenced by:  djudisj  5031