Theorem trpredeq1 9450

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

Assertion

Ref	Expression
trpredeq1	⊢ (𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Proof of Theorem trpredeq1

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

Step	Hyp	Ref	Expression
1		predeq1 6201	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
2	1	iuneq2d 4958	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦))
3	2	mpteq2dv 5180	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)))
4		predeq1 6201	. . . . . 6 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
5		rdgeq12 8228	. . . . . 6 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)) ∧ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋)) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)))
6	3, 4, 5	syl2anc 583	. . . . 5 ⊢ (𝑅 = 𝑆 → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)))
7	6	reseq1d 5887	. . . 4 ⊢ (𝑅 = 𝑆 → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)) ↾ ω))
8	7	rneqd 5844	. . 3 ⊢ (𝑅 = 𝑆 → ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)) ↾ ω))
9	8	unieqd 4858	. 2 ⊢ (𝑅 = 𝑆 → ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)) ↾ ω))
10		df-trpred 9448	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11		df-trpred 9448	. 2 ⊢ TrPred(𝑆, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)), Pred(𝑆, 𝐴, 𝑋)) ↾ ω)
12	9, 10, 11	3eqtr4g 2804	1 ⊢ (𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3430  ∪ cuni 4844  ∪ ciun 4929   ↦ cmpt 5161  ran crn 5589   ↾ cres 5590  Predcpred 6198  ωcom 7700  reccrdg 8224  TrPredctrpred 9447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-xp 5594  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-iota 6388  df-fv 6438  df-ov 7271  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-trpred 9448

This theorem is referenced by:  trpredeq1d  9453