Theorem dmxpss 5159

Description: The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Assertion

Ref	Expression
dmxpss	⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Proof of Theorem dmxpss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
2	1	eldm2 4921	. . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3		opelxp1 4753	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
4	3	exlimiv 1644	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
5	2, 4	sylbi 121	. 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
6	5	ssriv 3228	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:  ∃wex 1538   ∈ wcel 2200   ⊆ wss 3197  ⟨cop 3669   × cxp 4717  dom cdm 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-dm 4729

This theorem is referenced by:  rnxpss  5160  dmxpss2  5161  ssxpbm  5164  ssxp1  5165  funssxp  5495  tfrlemibfn  6480  tfr1onlembfn  6496  tfrcllembfn  6509  frecuzrdgtcl  10642  frecuzrdgdomlem  10647  dvbssntrcntop  15366