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Theorem trpredeq3 30966
 Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq3 (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Proof of Theorem trpredeq3
Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq3 5601 . . . . . 6 (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
2 rdgeq2 7395 . . . . . 6 (Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌) → rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)))
31, 2syl 17 . . . . 5 (𝑋 = 𝑌 → rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)))
43reseq1d 5316 . . . 4 (𝑋 = 𝑌 → (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
54rneqd 5274 . . 3 (𝑋 = 𝑌 → ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
65unieqd 4382 . 2 (𝑋 = 𝑌 ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
7 df-trpred 30962 . 2 TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
8 df-trpred 30962 . 2 TrPred(𝑅, 𝐴, 𝑌) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω)
96, 7, 83eqtr4g 2669 1 (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  Predcpred 5596  ωcom 6957  reccrdg 7392  TrPredctrpred 30961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-trpred 30962 This theorem is referenced by:  trpredeq3d  30969  dftrpred3g  30977
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