Theorem frrlem14 8324

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 8317	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → dom 𝐹 ⊆ 𝐴)
5		eldifn 4132	. . . . . . 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 8323	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ 𝐵)
15		elssuni 4937	. . . . . . . . . . 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 8315	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
18	16, 17	sseqtrrdi 4025	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ⊆ 𝐹)
19		dmss 5913	. . . . . . . . 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 4179	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
22		vsnid 4663	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
23	21, 22	sselii 3980	. . . . . . . . . 10 ⊢ 𝑧 ∈ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
24	8	dmeqi 5915	. . . . . . . . . . 11 ⊢ dom 𝐶 = dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
25		dmun 5921	. . . . . . . . . . 11 ⊢ dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
26		ovex 7464	. . . . . . . . . . . . 13 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
27	26	dmsnop 6236	. . . . . . . . . . . 12 ⊢ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
28	27	uneq2i 4165	. . . . . . . . . . 11 ⊢ (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
29	24, 25, 28	3eqtri 2769	. . . . . . . . . 10 ⊢ dom 𝐶 = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
30	23, 29	eleqtrri 2840	. . . . . . . . 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 3984	. . . . . . 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 3149	. . . 4 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
36		df-ne 2941	. . . . . 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 4366	. . 3 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
42	40, 41	sylibr 234	. 2 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
43	4, 42	eqssd 4001	1 ⊢ (𝜑 → dom 𝐹 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  ⟨cop 4632  ∪ cuni 4907   class class class wbr 5143   Fr wfr 5634  dom cdm 5685   ↾ cres 5687  Predcpred 6320   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  frecscfrecs 8305

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-frecs 8306

This theorem is referenced by:  fpr1  8328  frr1  9799