Theorem frrlem16 9795

Description: Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)

Assertion

Ref	Expression
frrlem16	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))

Distinct variable groups:   𝑤,𝑅,𝑧   𝑤,𝐴,𝑧

Proof of Theorem frrlem16

Step	Hyp	Ref	Expression
1		predres 6361	. . . . . 6 ⊢ Pred(𝑅, 𝐴, 𝑤) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑤)
2		relres 6025	. . . . . . . 8 ⊢ Rel (𝑅 ↾ 𝐴)
3		ssttrcl 9752	. . . . . . . 8 ⊢ (Rel (𝑅 ↾ 𝐴) → (𝑅 ↾ 𝐴) ⊆ t++(𝑅 ↾ 𝐴))
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (𝑅 ↾ 𝐴) ⊆ t++(𝑅 ↾ 𝐴)
5		predrelss 6359	. . . . . . 7 ⊢ ((𝑅 ↾ 𝐴) ⊆ t++(𝑅 ↾ 𝐴) → Pred((𝑅 ↾ 𝐴), 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑤))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Pred((𝑅 ↾ 𝐴), 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑤)
7	1, 6	eqsstri 4029	. . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑤)
8		inss1 4244	. . . . . . . . 9 ⊢ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ⊆ t++(𝑅 ↾ 𝐴)
9		coss1 5868	. . . . . . . . 9 ⊢ ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ⊆ t++(𝑅 ↾ 𝐴) → ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ (t++(𝑅 ↾ 𝐴) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ (t++(𝑅 ↾ 𝐴) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)))
11		coss2 5869	. . . . . . . . 9 ⊢ ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ⊆ t++(𝑅 ↾ 𝐴) → (t++(𝑅 ↾ 𝐴) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ (t++(𝑅 ↾ 𝐴) ∘ t++(𝑅 ↾ 𝐴)))
12	8, 11	ax-mp 5	. . . . . . . 8 ⊢ (t++(𝑅 ↾ 𝐴) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ (t++(𝑅 ↾ 𝐴) ∘ t++(𝑅 ↾ 𝐴))
13	10, 12	sstri 4004	. . . . . . 7 ⊢ ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ (t++(𝑅 ↾ 𝐴) ∘ t++(𝑅 ↾ 𝐴))
14		ttrcltr 9753	. . . . . . 7 ⊢ (t++(𝑅 ↾ 𝐴) ∘ t++(𝑅 ↾ 𝐴)) ⊆ t++(𝑅 ↾ 𝐴)
15	13, 14	sstri 4004	. . . . . 6 ⊢ ((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ t++(𝑅 ↾ 𝐴)
16		predtrss 6344	. . . . . 6 ⊢ ((((t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴)) ∘ (t++(𝑅 ↾ 𝐴) ∩ (𝐴 × 𝐴))) ⊆ t++(𝑅 ↾ 𝐴) ∧ 𝑤 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧) ∧ 𝑧 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
17	15, 16	mp3an1 1447	. . . . 5 ⊢ ((𝑤 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧) ∧ 𝑧 ∈ 𝐴) → Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
18	7, 17	sstrid 4006	. . . 4 ⊢ ((𝑤 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧) ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
19	18	ancoms 458	. . 3 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)) → Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
20	19	ralrimiva 3143	. 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
21	20	adantl 481	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ∀wral 3058   ∩ cin 3961   ⊆ wss 3962   Fr wfr 5637   Se wse 5638   × cxp 5686   ↾ cres 5690   ∘ ccom 5692  Rel wrel 5693  Predcpred 6321  t++cttrcl 9744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-ttrcl 9745

This theorem is referenced by:  frr1  9796