Theorem dmxpss 5061

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 2742	. . . 4 ⊢ 𝑥 ∈ V
2	1	eldm2 4827	. . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3		opelxp1 4662	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
4	3	exlimiv 1598	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
5	2, 4	sylbi 121	. 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴)
6	5	ssriv 3161	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:  ∃wex 1492   ∈ wcel 2148   ⊆ wss 3131  ⟨cop 3597   × cxp 4626  dom cdm 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-dm 4638

This theorem is referenced by:  rnxpss  5062  dmxpss2  5063  ssxpbm  5066  ssxp1  5067  funssxp  5387  tfrlemibfn  6331  tfr1onlembfn  6347  tfrcllembfn  6360  frecuzrdgtcl  10414  frecuzrdgdomlem  10419  dvbssntrcntop  14238