Theorem trpredpred 9406

Description: Assuming it is a set, the predecessor class is a subset of the class of transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)

Assertion

Ref	Expression
trpredpred	⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋))

Proof of Theorem trpredpred

Dummy variables 𝑎 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fr0g 8237	. . . . . 6 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
2		frfnom 8236	. . . . . . 7 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) Fn ω
3		peano1 7710	. . . . . . 7 ⊢ ∅ ∈ ω
4		fnbrfvb 6804	. . . . . . 7 ⊢ (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) Fn ω ∧ ∅ ∈ ω) → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋) ↔ ∅(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋)))
5	2, 3, 4	mp2an 688	. . . . . 6 ⊢ (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋) ↔ ∅(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋))
6	1, 5	sylib 217	. . . . 5 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ∅(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋))
7		0ex 5226	. . . . . 6 ⊢ ∅ ∈ V
8		breq1 5073	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑧(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋) ↔ ∅(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋)))
9	7, 8	spcev 3535	. . . . 5 ⊢ (∅(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋) → ∃𝑧 𝑧(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋))
10	6, 9	syl 17	. . . 4 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ∃𝑧 𝑧(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋))
11		elrng 5789	. . . 4 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → (Pred(𝑅, 𝐴, 𝑋) ∈ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) ↔ ∃𝑧 𝑧(rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)Pred(𝑅, 𝐴, 𝑋)))
12	10, 11	mpbird 256	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ∈ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
13		elssuni 4868	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
14	12, 13	syl 17	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
15		df-trpred 9396	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
16	14, 15	sseqtrrdi 3968	1 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153  ran crn 5581   ↾ cres 5582  Predcpred 6190   Fn wfn 6413  ‘cfv 6418  ωcom 7687  reccrdg 8211  TrPredctrpred 9395

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-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-trpred 9396

This theorem is referenced by:  dftrpred3g  9412  trpredpo  9414  frmin  9438  frr1  9448