Theorem wfrlem16OLD 8126

Description: Lemma for well-ordered recursion. If 𝑧 is 𝑅 minimal in (𝐴 ∖ dom 𝐹), then 𝐶 is acceptable and thus a subset of 𝐹, but dom 𝐶 is bigger than dom 𝐹. Thus, 𝑧 cannot be minimal, so (𝐴 ∖ dom 𝐹) must be empty, and (due to wfrdmssOLD 8117), dom 𝐹 = 𝐴. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13OLD.1	⊢ 𝑅 We 𝐴
wfrlem13OLD.2	⊢ 𝑅 Se 𝐴
wfrlem13OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem16OLD	⊢ dom 𝐹 = 𝐴

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

Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem16OLD

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem13OLD.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2	1	wfrdmssOLD 8117	. 2 ⊢ dom 𝐹 ⊆ 𝐴
3		eldifn 4058	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
4		ssun2 4103	. . . . . . . . 9 ⊢ {𝑧} ⊆ (dom 𝐹 ∪ {𝑧})
5		vsnid 4595	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
6	4, 5	sselii 3914	. . . . . . . 8 ⊢ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
7		wfrlem13OLD.4	. . . . . . . . . 10 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
8	7	dmeqi 5802	. . . . . . . . 9 ⊢ dom 𝐶 = dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9		dmun 5808	. . . . . . . . 9 ⊢ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
10		fvex 6769	. . . . . . . . . . 11 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
11	10	dmsnop 6108	. . . . . . . . . 10 ⊢ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
12	11	uneq2i 4090	. . . . . . . . 9 ⊢ (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
13	8, 9, 12	3eqtri 2770	. . . . . . . 8 ⊢ dom 𝐶 = (dom 𝐹 ∪ {𝑧})
14	6, 13	eleqtrri 2838	. . . . . . 7 ⊢ 𝑧 ∈ dom 𝐶
15		wfrlem13OLD.1	. . . . . . . . . . . 12 ⊢ 𝑅 We 𝐴
16		wfrlem13OLD.2	. . . . . . . . . . . 12 ⊢ 𝑅 Se 𝐴
17	15, 16, 1, 7	wfrlem15OLD 8125	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
18		elssuni 4868	. . . . . . . . . . 11 ⊢ (𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝐶 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
20		dfwrecsOLD 8100	. . . . . . . . . . 11 ⊢ wrecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21	1, 20	eqtri 2766	. . . . . . . . . 10 ⊢ 𝐹 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
22	19, 21	sseqtrrdi 3968	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ⊆ 𝐹)
23		dmss 5800	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐹 → dom 𝐶 ⊆ dom 𝐹)
24	22, 23	syl 17	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐶 ⊆ dom 𝐹)
25	24	sseld 3916	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑧 ∈ dom 𝐶 → 𝑧 ∈ dom 𝐹))
26	14, 25	mpi 20	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝑧 ∈ dom 𝐹)
27	3, 26	mtand 812	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
28	27	nrex 3196	. . . 4 ⊢ ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅
29		df-ne 2943	. . . . 5 ⊢ ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
30		difss 4062	. . . . . 6 ⊢ (𝐴 ∖ dom 𝐹) ⊆ 𝐴
31	15, 16	tz6.26i 6237	. . . . . 6 ⊢ (((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
32	30, 31	mpan 686	. . . . 5 ⊢ ((𝐴 ∖ dom 𝐹) ≠ ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
33	29, 32	sylbir 234	. . . 4 ⊢ (¬ (𝐴 ∖ dom 𝐹) = ∅ → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
34	28, 33	mt3 200	. . 3 ⊢ (𝐴 ∖ dom 𝐹) = ∅
35		ssdif0 4294	. . 3 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
36	34, 35	mpbir 230	. 2 ⊢ 𝐴 ⊆ dom 𝐹
37	2, 36	eqssi 3933	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   Se wse 5533   We wwe 5534  dom cdm 5580   ↾ cres 5582  Predcpred 6190   Fn wfn 6413  ‘cfv 6418  wrecscwrecs 8098

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-3or 1086  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-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  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-2nd 7805  df-frecs 8068  df-wrecs 8099

This theorem is referenced by:  wfr1OLD  8129  wfr2OLD  8130