Theorem rnxpss2 5054

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

Assertion

Ref	Expression
rnxpss2	⊢ (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅 ⊆ 𝐵)

Proof of Theorem rnxpss2

Step	Hyp	Ref	Expression
1		rnss 4850	. 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅 ⊆ ran (𝐴 × 𝐵))
2		rnxpss 5052	. 2 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
3	1, 2	sstrdi 3165	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅 ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3127   × cxp 4618  ran crn 4621

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by:  cossxp2  5144