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Theorem trpredeq2 33174
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.
StepHypRef Expression
1 predeq2 6123 . . . . . . 7 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐵, 𝑦))
21iuneq2d 4913 . . . . . 6 (𝐴 = 𝐵 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦))
32mpteq2dv 5129 . . . . 5 (𝐴 = 𝐵 → (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)))
4 predeq2 6123 . . . . 5 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
5 rdgeq12 8036 . . . . . 6 (((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)) ∧ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) → rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)))
65reseq1d 5821 . . . . 5 (((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)) ∧ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) → (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
73, 4, 6syl2anc 587 . . . 4 (𝐴 = 𝐵 → (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
87rneqd 5776 . . 3 (𝐴 = 𝐵 → ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
98unieqd 4817 . 2 (𝐴 = 𝐵 ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω))
10 df-trpred 33171 . 2 TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11 df-trpred 33171 . 2 TrPred(𝑅, 𝐵, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω)
129, 10, 113eqtr4g 2861 1 (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  Vcvv 3444   cuni 4803   ciun 4884  cmpt 5113  ran crn 5524  cres 5525  Predcpred 6119  ωcom 7564  reccrdg 8032  TrPredctrpred 33170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-iota 6287  df-fv 6336  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-trpred 33171
This theorem is referenced by:  trpredeq2d  33177
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