Theorem setlikespec 6330

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 3419	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)}
2		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpred 6323	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)))
4	3	eqabdv 2898	. . . 4 ⊢ (𝑋 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)})
5	1, 4	eqtr4id 2819	. . 3 ⊢ (𝑋 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
6	5	adantr 486	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = Pred(𝑅, 𝐴, 𝑋))
7		seex 5622	. . 3 ⊢ ((𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} ∈ V)
8	7	ancoms 464	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} ∈ V)
9	6, 8	eqeltrrd 2866	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  {crab 3418  Vcvv 3457   class class class wbr 5111   Se wse 5614  Predcpred 6305

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-se 5617  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306

This theorem is used by:  sexp2  8144  sexp3  8151  fpr1  8302  ttrclselem2  9698  frmin  9724  frr1  9734