Theorem dfpred2 6302

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 6292	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		dfpred2.1	. . . 4 ⊢ 𝑋 ∈ V
3		iniseg 6090	. . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
4	2, 3	ax-mp 5	. . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}
5	4	ineq2i 4172	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
6	1, 5	eqtri 2788	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})

Syntax hints:   = wceq 1563   ∈ wcel 2145  {cab 2743  Vcvv 3457   ∩ cin 3906  {csn 4585   class class class wbr 5105  ^◡ccnv 5651   “ cima 5655  Predcpred 6291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292

This theorem is referenced by:  dfpred3  6303