Theorem dfpred3g 6344

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 6336	. . 3 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2		breq2 5170	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋))
3	2	rabbidv 3451	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
4	1, 3	eqeq12d 2756	. 2 ⊢ (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
5		vex 3492	. . 3 ⊢ 𝑥 ∈ V
6	5	dfpred3 6343	. 2 ⊢ Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
7	4, 6	vtoclg 3566	1 ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443   class class class wbr 5166  Predcpred 6331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332

This theorem is referenced by:  lrrecpred  27995  fnrelpredd  35065  wsuclem  35789