Theorem dfpred3g 6265

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 6257	. . 3 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2		breq2 5097	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋))
3	2	rabbidv 3403	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
4	1, 3	eqeq12d 2749	. 2 ⊢ (𝑥 = 𝑋 → (Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ↔ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}))
5		vex 3441	. . 3 ⊢ 𝑥 ∈ V
6	5	dfpred3 6264	. 2 ⊢ Pred(𝑅, 𝐴, 𝑥) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥}
7	4, 6	vtoclg 3508	1 ⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3396   class class class wbr 5093  Predcpred 6252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253

This theorem is referenced by:  lrrecpred  27888  fnrelpredd  35123  wsuclem  35888  dfpred4  38512