Theorem dfpred3 6341

Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Hypothesis

Ref	Expression
dfpred2.1	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
dfpred3	⊢ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}

Distinct variable groups:   𝑦,𝑅   𝑦,𝑋   𝑦,𝐴

Proof of Theorem dfpred3

Step	Hyp	Ref	Expression
1		incom 4109	. 2 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
2		dfpred2.1	. . 3 ⊢ 𝑋 ∈ V
3	2	dfpred2 6340	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
4		dfrab2 4268	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
5	1, 3, 4	3eqtr4i 2796	1 ⊢ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741  {crab 3416  Vcvv 3454   ∩ cin 4092   class class class wbr 5108  Predcpred 6329

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330

This theorem is used by:  dfpred3g  6342  frpomin2  6370  fnrelpredd  35845  nummin  35847