Theorem dfpred3 6321

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 4108	. 2 ⊢ (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋}) = ({𝑦 ∣ 𝑦𝑅𝑋} ∩ 𝐴)
2		dfpred2.1	. . 3 ⊢ 𝑋 ∈ V
3	2	dfpred2 6320	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
4		dfrab2 4267	. 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 1569   ∈ wcel 2146  {cab 2741  {crab 3415  Vcvv 3453   ∩ cin 4091   class class class wbr 5105  Predcpred 6309

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310

This theorem is used by:  dfpred3g  6322  frpomin2  6350  fnrelpredd  35781  nummin  35783