Theorem dfpred2 6258

Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Hypothesis

Ref	Expression
dfpred2.1	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
dfpred2	⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})

Distinct variable groups:   𝑦,𝑅   𝑦,𝑋

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfpred2

Step	Hyp	Ref	Expression
1		df-pred 6248	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		dfpred2.1	. . . 4 ⊢ 𝑋 ∈ V
3		iniseg 6046	. . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
4	2, 3	ax-mp 5	. . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}
5	4	ineq2i 4167	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
6	1, 5	eqtri 2754	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})

Syntax hints:   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   ∩ cin 3901  {csn 4576   class class class wbr 5091  ^◡ccnv 5615   “ cima 5619  Predcpred 6247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248

This theorem is referenced by:  dfpred3  6259