Theorem dfpred2 6309

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 6299	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		dfpred2.1	. . . 4 ⊢ 𝑋 ∈ V
3		iniseg 6093	. . . 4 ⊢ (𝑋 ∈ V → (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋})
4	2, 3	ax-mp 5	. . 3 ⊢ (◡𝑅 “ {𝑋}) = {𝑦 ∣ 𝑦𝑅𝑋}
5	4	ineq2i 4163	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
6	1, 5	eqtri 2783	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∩ cin 3898  {csn 4584   class class class wbr 5103  ^◡ccnv 5654   “ cima 5658  Predcpred 6298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299

This theorem is used by:  dfpred3  6310