Theorem trpredeq2 9315

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 6152	. . . . . . 7 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐵, 𝑦))
2	1	iuneq2d 4923	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦))
3	2	mpteq2dv 5140	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐵, 𝑦)))
4		predeq2 6152	. . . . 5 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
5		rdgeq12 8138	. . . . . 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 5839	. . . . 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 587	. . . 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 5796	. . 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 4823	. 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 9312	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11		df-trpred 9312	. 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 2799	1 ⊢ (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  Vcvv 3401  ∪ cuni 4809  ∪ ciun 4894   ↦ cmpt 5124  ran crn 5541   ↾ cres 5542  Predcpred 6148  ωcom 7633  reccrdg 8134  TrPredctrpred 9311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-xp 5546  df-cnv 5548  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-iota 6327  df-fv 6377  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-trpred 9312

This theorem is referenced by:  trpredeq2d  9318