Theorem frrlem14 8303

Description: Lemma for well-founded recursion. Finally, we tie all these threads together and show that dom 𝐹 = 𝐴 when given the right 𝑆. Specifically, we prove that there can be no 𝑅-minimal element of (𝐴 ∖ dom 𝐹). (Contributed by Scott Fenton, 7-Dec-2022.)

Hypotheses

Ref	Expression
frrlem11.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem11.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem11.3	⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
frrlem11.4	⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
frrlem12.5	⊢ (𝜑 → 𝑅 Fr 𝐴)
frrlem12.6	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
frrlem12.7	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
frrlem13.8	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ∈ V)
frrlem13.9	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ⊆ 𝐴)
frrlem14.10	⊢ ((𝜑 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)

Assertion

Ref	Expression
frrlem14	⊢ (𝜑 → dom 𝐹 = 𝐴)

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧   𝑅,𝑓,𝑥,𝑦,𝑧   𝐵,𝑔,ℎ,𝑧   𝑥,𝐹,𝑢,𝑣,𝑧   𝜑,𝑓,𝑧   𝑓,𝐹   𝜑,𝑔,ℎ,𝑥,𝑢,𝑣   𝐴,ℎ,𝑤,𝑓,𝑦,𝑥   𝑤,𝐺   𝑤,𝑅   𝑦,𝐹   𝑥,𝐵   𝑤,𝐶   𝑤,𝐹   𝜑,𝑤   𝑤,𝑆   𝑧,𝑤

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔)   𝐵(𝑦,𝑤,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,ℎ)   𝑅(𝑣,𝑢,𝑔,ℎ)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,ℎ)   𝐹(𝑔,ℎ)   𝐺(𝑣,𝑢,𝑔,ℎ)

Proof of Theorem frrlem14

Step	Hyp	Ref	Expression
1		frrlem11.1	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2		frrlem11.2	. . . 4 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3	1, 2	frrlem7 8296	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → dom 𝐹 ⊆ 𝐴)
5		eldifn 4112	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ 𝑧 ∈ dom 𝐹)
7		frrlem11.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
8		frrlem11.4	. . . . . . . . . . . 12 ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9		frrlem12.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 Fr 𝐴)
10		frrlem12.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆)
11		frrlem12.7	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)
12		frrlem13.8	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ∈ V)
13		frrlem13.9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ⊆ 𝐴)
14	1, 2, 7, 8, 9, 10, 11, 12, 13	frrlem13 8302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ 𝐵)
15		elssuni 4918	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ⊆ ∪ 𝐵)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ⊆ ∪ 𝐵)
17	1, 2	frrlem5 8294	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
18	16, 17	sseqtrrdi 4005	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ⊆ 𝐹)
19		dmss 5887	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐹 → dom 𝐶 ⊆ dom 𝐹)
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → dom 𝐶 ⊆ dom 𝐹)
21		ssun2 4159	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
22		vsnid 4644	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
23	21, 22	sselii 3960	. . . . . . . . . 10 ⊢ 𝑧 ∈ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
24	8	dmeqi 5889	. . . . . . . . . . 11 ⊢ dom 𝐶 = dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
25		dmun 5895	. . . . . . . . . . 11 ⊢ dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
26		ovex 7443	. . . . . . . . . . . . 13 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
27	26	dmsnop 6210	. . . . . . . . . . . 12 ⊢ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
28	27	uneq2i 4145	. . . . . . . . . . 11 ⊢ (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
29	24, 25, 28	3eqtri 2763	. . . . . . . . . 10 ⊢ dom 𝐶 = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
30	23, 29	eleqtrri 2834	. . . . . . . . 9 ⊢ 𝑧 ∈ dom 𝐶
31	30	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐶)
32	20, 31	sseldd 3964	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ dom 𝐹)
33	32	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → 𝑧 ∈ dom 𝐹))
34	6, 33	mtod 198	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
35	34	nrexdv 3136	. . . 4 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
36		df-ne 2934	. . . . . 6 ⊢ ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
37		frrlem14.10	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
38	36, 37	sylan2br 595	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐴 ∖ dom 𝐹) = ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
39	38	ex 412	. . . 4 ⊢ (𝜑 → (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅))
40	35, 39	mt3d 148	. . 3 ⊢ (𝜑 → (𝐴 ∖ dom 𝐹) = ∅)
41		ssdif0 4346	. . 3 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
42	40, 41	sylibr 234	. 2 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
43	4, 42	eqssd 3981	1 ⊢ (𝜑 → dom 𝐹 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612  ∪ cuni 4888   class class class wbr 5124   Fr wfr 5608  dom cdm 5659   ↾ cres 5661  Predcpred 6294   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  frecscfrecs 8284

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-fr 5611  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-frecs 8285

This theorem is referenced by:  fpr1  8307  frr1  9778