Theorem trpredeq1 32316

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 5937	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
2	1	iuneq2d 4782	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦))
3	2	mpteq2dv 4982	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)))
4		predeq1 5937	. . . . . 6 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
5		rdgeq12 7794	. . . . . 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 579	. . . . 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 5643	. . . 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 5600	. . 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 4683	. 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 32314	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11		df-trpred 32314	. 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 2839	1 ⊢ (𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1601  Vcvv 3398  ∪ cuni 4673  ∪ ciun 4755   ↦ cmpt 4967  ran crn 5358   ↾ cres 5359  Predcpred 5934  ωcom 7345  reccrdg 7790  TrPredctrpred 32313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-xp 5363  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-iota 6101  df-fv 6145  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-trpred 32314

This theorem is referenced by:  trpredeq1d  32319