Theorem setlikespec 6272

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 3396	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)}
2		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpred 6265	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)))
4	3	eqabdv 2864	. . . 4 ⊢ (𝑋 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)})
5	1, 4	eqtr4id 2785	. . 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 5575	. . 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 2832	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  {crab 3395  Vcvv 3436   class class class wbr 5091   Se wse 5567  Predcpred 6247

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-se 5570  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248

This theorem is referenced by:  sexp2  8076  sexp3  8083  fpr1  8233  ttrclselem2  9616  frmin  9639  frr1  9649