Theorem trpredeq2 9399

Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
trpredeq2	⊢ (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Proof of Theorem trpredeq2

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

Step	Hyp	Ref	Expression
1		predeq2 6194	. . . . . . 7 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐵, 𝑦))
2	1	iuneq2d 4950	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦))
3	2	mpteq2dv 5172	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)))
4		predeq2 6194	. . . . 5 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
5		rdgeq12 8215	. . . . . 6 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)) ∧ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)))
6	5	reseq1d 5879	. . . . 5 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)) ∧ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
7	3, 4, 6	syl2anc 583	. . . 4 ⊢ (𝐴 = 𝐵 → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
8	7	rneqd 5836	. . 3 ⊢ (𝐴 = 𝐵 → ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
9	8	unieqd 4850	. 2 ⊢ (𝐴 = 𝐵 → ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
10		df-trpred 9396	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11		df-trpred 9396	. 2 ⊢ TrPred(𝑅, 𝐵, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω)
12	9, 10, 11	3eqtr4g 2804	1 ⊢ (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  Vcvv 3422  ∪ cuni 4836  ∪ ciun 4921   ↦ cmpt 5153  ran crn 5581   ↾ cres 5582  Predcpred 6190  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-ext 2709

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-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fv 6426  df-ov 7258  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-trpred 9396

This theorem is referenced by:  trpredeq2d  9402