Theorem dmxpss2 5171

Description: Upper bound for the domain of a binary relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
dmxpss2	⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴)

Proof of Theorem dmxpss2

Step	Hyp	Ref	Expression
1		dmss 4932	. 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ dom (𝐴 × 𝐵))
2		dmxpss 5169	. 2 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
3	1, 2	sstrdi 3238	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3199   × cxp 4725  dom cdm 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-xp 4733  df-dm 4737

This theorem is referenced by:  cossxp2  5262