Theorem trpredeq3 33085

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.

Step	Hyp	Ref	Expression
1		predeq3 6149	. . . . . 6 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
2		rdgeq2 8045	. . . . . 6 ⊢ (Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)))
3	1, 2	syl 17	. . . . 5 ⊢ (𝑋 = 𝑌 → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)))
4	3	reseq1d 5849	. . . 4 ⊢ (𝑋 = 𝑌 → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
5	4	rneqd 5805	. . 3 ⊢ (𝑋 = 𝑌 → ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
6	5	unieqd 4849	. 2 ⊢ (𝑋 = 𝑌 → ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω))
7		df-trpred 33081	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
8		df-trpred 33081	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑌) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω)
9	6, 7, 8	3eqtr4g 2880	1 ⊢ (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1536  Vcvv 3493  ∪ cuni 4835  ∪ ciun 4916   ↦ cmpt 5143  ran crn 5553   ↾ cres 5554  Predcpred 6144  ωcom 7577  reccrdg 8042  TrPredctrpred 33080

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-rab 3146  df-v 3495  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5558  df-cnv 5560  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-iota 6311  df-fv 6360  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-trpred 33081

This theorem is referenced by:  trpredeq3d  33088  dftrpred3g  33096