Theorem frrlem14 8086

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 8079	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → dom 𝐹 ⊆ 𝐴)
5		eldifn 4058	. . . . . . 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 8085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ 𝐵)
15		elssuni 4868	. . . . . . . . . . 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 8077	. . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵
18	16, 17	sseqtrrdi 3968	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ⊆ 𝐹)
19		dmss 5800	. . . . . . . . 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 4103	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
22		vsnid 4595	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
23	21, 22	sselii 3914	. . . . . . . . . 10 ⊢ 𝑧 ∈ (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
24	8	dmeqi 5802	. . . . . . . . . . 11 ⊢ dom 𝐶 = dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
25		dmun 5808	. . . . . . . . . . 11 ⊢ dom ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
26		ovex 7288	. . . . . . . . . . . . 13 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
27	26	dmsnop 6108	. . . . . . . . . . . 12 ⊢ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
28	27	uneq2i 4090	. . . . . . . . . . 11 ⊢ (dom (𝐹 ↾ 𝑆) ∪ dom {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
29	24, 25, 28	3eqtri 2770	. . . . . . . . . 10 ⊢ dom 𝐶 = (dom (𝐹 ↾ 𝑆) ∪ {𝑧})
30	23, 29	eleqtrri 2838	. . . . . . . . 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 3918	. . . . . . 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 197	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
35	34	nrexdv 3197	. . . 4 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
36		df-ne 2943	. . . . . 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 594	. . . . 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 4294	. . 3 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
42	40, 41	sylibr 233	. 2 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
43	4, 42	eqssd 3934	1 ⊢ (𝜑 → dom 𝐹 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070   Fr wfr 5532  dom cdm 5580   ↾ cres 5582  Predcpred 6190   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  frecscfrecs 8067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-fr 5535  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-frecs 8068

This theorem is referenced by:  fpr1  8090  frr1  9448