Theorem dmxpss 5034

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 2729	. . . 4 ⊢ 𝑥 ∈ V
2	1	eldm2 4802	. . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3		opelxp1 4638	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
4	3	exlimiv 1586	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
5	2, 4	sylbi 120	. 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
6	5	ssriv 3146	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:  ∃wex 1480   ∈ wcel 2136   ⊆ wss 3116  ⟨cop 3579   × cxp 4602  dom cdm 4604

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  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-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-dm 4614

This theorem is referenced by:  rnxpss  5035  dmxpss2  5036  ssxpbm  5039  ssxp1  5040  funssxp  5357  tfrlemibfn  6296  tfr1onlembfn  6312  tfrcllembfn  6325  frecuzrdgtcl  10347  frecuzrdgdomlem  10352  dvbssntrcntop  13293