Theorem predss 5590

Description: The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predss	⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Proof of Theorem predss

Step	Hyp	Ref	Expression
1		df-pred 5583	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 3794	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3597	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Syntax hints:   ∩ cin 3538   ⊆ wss 3539  {csn 4124  ^◡ccnv 5027   “ cima 5031  Predcpred 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-in 3546  df-ss 3553  df-pred 5583

This theorem is referenced by:  wfr3g  7278  wfrlem4  7283  wfrlem10  7289  trpredlem1  30805  wsuclem  30851  wsuclemOLD  30852  frr3g  30857  frrlem4  30861