Theorem setlikespec 6217

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 3072	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑋} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)}
2		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpred 6208	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)))
4	3	abbi2dv 2876	. . . 4 ⊢ (𝑋 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑋) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑋)})
5	1, 4	eqtr4id 2798	. . 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 5542	. . 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 2840	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422   class class class wbr 5070   Se wse 5533  Predcpred 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-se 5536  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191

This theorem is referenced by:  fpr1  8090  wfrlem15OLD  8125  trpredtr  9408  trpredmintr  9409  trpredelss  9411  dftrpred3g  9412  trpredpo  9414  trpredrec  9415  frmin  9438  frr1  9448  ttrclselem2  33712  sexp2  33720  sexp3  33726