Theorem dfpred3g 6316

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

Assertion

Ref	Expression
dfpred3g	⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})

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

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfpred3g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		predeq3 6308	. . 3 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2		breq2 5107	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋))
3	2	rabbidv 3420	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
4	1, 3	eqeq12d 2777	. 2 ⊢ (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
5		vex 3455	. . 3 ⊢ 𝑥 ∈ V
6	5	dfpred3 6315	. 2 ⊢ Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
7	4, 6	vtoclg 3518	1 ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3413   class class class wbr 5103  Predcpred 6303

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 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304

This theorem is used by:  lrrecpred  28330  fnrelpredd  35720  wsuclem  36587  dfpred4  39411