Theorem dmxpss 5041

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 2733	. . . 4 ⊢ 𝑥 ∈ V
2	1	eldm2 4809	. . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3		opelxp1 4645	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
4	3	exlimiv 1591	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
5	2, 4	sylbi 120	. 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
6	5	ssriv 3151	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:  ∃wex 1485   ∈ wcel 2141   ⊆ wss 3121  ⟨cop 3586   × cxp 4609  dom cdm 4611

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621

This theorem is referenced by:  rnxpss  5042  dmxpss2  5043  ssxpbm  5046  ssxp1  5047  funssxp  5367  tfrlemibfn  6307  tfr1onlembfn  6323  tfrcllembfn  6336  frecuzrdgtcl  10368  frecuzrdgdomlem  10373  dvbssntrcntop  13447