Theorem bnj1029 35265

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1029	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Proof of Theorem bnj1029

Dummy variables 𝑓 𝑖 𝑚 𝑛 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 263	. 2 ⊢ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		biid 263	. 2 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		biid 263	. 2 ⊢ ((𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4		biid 263	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5		biid 263	. 2 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
6		biid 263	. 2 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
7		biid 263	. 2 ⊢ ([𝑝 / 𝑛](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [𝑝 / 𝑛](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8		biid 263	. 2 ⊢ ([𝑝 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑝 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
9		biid 263	. 2 ⊢ ([𝑝 / 𝑛](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑝 / 𝑛](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
10		biid 263	. 2 ⊢ ([(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ [(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
11		biid 263	. 2 ⊢ ([(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12		biid 263	. 2 ⊢ ([(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [(𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓][𝑝 / 𝑛](𝑛 ∈ (ω ∖ {∅}) ∧ 𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
13		eqid 2764	. 2 ⊢ (ω ∖ {∅}) = (ω ∖ {∅})
14		eqid 2764	. 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))} = {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}
15		eqid 2764	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
16		eqid 2764	. 2 ⊢ (𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑛, ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)⟩})
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	bnj907 35264	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088  [wsbc 3746   ∖ cdif 3903   ∪ cun 3904  ∅c0 4287  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848   ∧ w-bnj17 34984   predc-bnj14 34986   FrSe w-bnj15 34990   trClc-bnj18 34992   TrFow-bnj19 34994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-res 5661  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-om 7849  df-bnj17 34985  df-bnj14 34987  df-bnj13 34989  df-bnj15 34991  df-bnj18 34993  df-bnj19 34995

This theorem is referenced by:  bnj1125  35289