Theorem dfpred2 6201

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 6191	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		dfpred2.1	. . . 4 ⊢ 𝑋 ∈ V
3		iniseg 5994	. . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
4	2, 3	ax-mp 5	. . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}
5	4	ineq2i 4140	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
6	1, 5	eqtri 2766	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ∩ cin 3882  {csn 4558   class class class wbr 5070  ^◡ccnv 5579   “ cima 5583  Predcpred 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191

This theorem is referenced by:  dfpred3  6202  tz6.26OLD  6236