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Theorem trpredeq2 31414
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 5645 . . . . . . 7 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐵, 𝑦))
21iuneq2d 4518 . . . . . 6 (𝐴 = 𝐵 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦))
32mpteq2dv 4710 . . . . 5 (𝐴 = 𝐵 → (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)))
4 predeq2 5645 . . . . 5 (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
5 rdgeq12 7455 . . . . . 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 5359 . . . . 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 692 . . . 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 5317 . . 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 4417 . 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 31411 . 2 TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11 df-trpred 31411 . 2 TrPred(𝑅, 𝐵, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐵, 𝑦)), Pred(𝑅, 𝐵, 𝑋)) ↾ ω)
129, 10, 113eqtr4g 2685 1 (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  Vcvv 3191   cuni 4407   ciun 4490  cmpt 4678  ran crn 5080  cres 5081  Predcpred 5641  ωcom 7013  reccrdg 7451  TrPredctrpred 31410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-iota 5813  df-fv 5858  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-trpred 31411
This theorem is referenced by:  trpredeq2d  31417
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