Theorem trpredeq1 33080

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 6143	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
2	1	iuneq2d 4941	. . . . . . 7 ⊢ (𝑅 = 𝑆 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦))
3	2	mpteq2dv 5155	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑆, 𝐴, 𝑦)))
4		predeq1 6143	. . . . . 6 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
5		rdgeq12 8042	. . . . . 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 586	. . . . 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 5845	. . . 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 5801	. . 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 4845	. 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 33078	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
11		df-trpred 33078	. 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 2880	1 ⊢ (𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1536  Vcvv 3491  ∪ cuni 4831  ∪ ciun 4912   ↦ cmpt 5139  ran crn 5549   ↾ cres 5550  Predcpred 6140  ωcom 7573  reccrdg 8038  TrPredctrpred 33077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-trpred 33078

This theorem is referenced by:  trpredeq1d  33083