Theorem setlikespec 6346

Description: If 𝑅 is set-like in 𝐴, then all predecessor classes of elements of 𝐴 exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
setlikespec	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem setlikespec

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3437	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)}
2		vex 3484	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpred 6338	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)))
4	3	eqabdv 2875	. . . 4 ⊢ (𝑋 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)})
5	1, 4	eqtr4id 2796	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
6	5	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7		seex 5644	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} ∈ V)
8	7	ancoms 458	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} ∈ V)
9	6, 8	eqeltrrd 2842	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  {crab 3436  Vcvv 3480   class class class wbr 5143   Se wse 5635  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-se 5638  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321

This theorem is referenced by:  sexp2  8171  sexp3  8178  fpr1  8328  wfrlem15OLD  8363  ttrclselem2  9766  frmin  9789  frr1  9799