Theorem wfrlem16OLD 8320

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 8311), 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 8311	. 2 ⊢ dom 𝐹 ⊆ 𝐴
3		eldifn 4126	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
4		ssun2 4172	. . . . . . . . 9 ⊢ {𝑧} ⊆ (dom 𝐹 ∪ {𝑧})
5		vsnid 4664	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
6	4, 5	sselii 3978	. . . . . . . 8 ⊢ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
7		wfrlem13OLD.4	. . . . . . . . . 10 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
8	7	dmeqi 5902	. . . . . . . . 9 ⊢ dom 𝐶 = dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9		dmun 5908	. . . . . . . . 9 ⊢ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
10		fvex 6901	. . . . . . . . . . 11 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
11	10	dmsnop 6212	. . . . . . . . . 10 ⊢ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
12	11	uneq2i 4159	. . . . . . . . 9 ⊢ (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
13	8, 9, 12	3eqtri 2764	. . . . . . . 8 ⊢ dom 𝐶 = (dom 𝐹 ∪ {𝑧})
14	6, 13	eleqtrri 2832	. . . . . . 7 ⊢ 𝑧 ∈ dom 𝐶
15		wfrlem13OLD.1	. . . . . . . . . . . 12 ⊢ 𝑅 We 𝐴
16		wfrlem13OLD.2	. . . . . . . . . . . 12 ⊢ 𝑅 Se 𝐴
17	15, 16, 1, 7	wfrlem15OLD 8319	. . . . . . . . . . 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 4940	. . . . . . . . . . 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 8294	. . . . . . . . . . 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 2760	. . . . . . . . . 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 4032	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ⊆ 𝐹)
23		dmss 5900	. . . . . . . . 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 3980	. . . . . . 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 814	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
28	27	nrex 3074	. . . 4 ⊢ ¬ ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅
29		df-ne 2941	. . . . 5 ⊢ ((𝐴 ∖ dom 𝐹) ≠ ∅ ↔ ¬ (𝐴 ∖ dom 𝐹) = ∅)
30		difss 4130	. . . . . 6 ⊢ (𝐴 ∖ dom 𝐹) ⊆ 𝐴
31	15, 16	tz6.26i 6347	. . . . . 6 ⊢ (((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
32	30, 31	mpan 688	. . . . 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 4362	. . 3 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (𝐴 ∖ dom 𝐹) = ∅)
36	34, 35	mpbir 230	. 2 ⊢ 𝐴 ⊆ dom 𝐹
37	2, 36	eqssi 3997	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  {csn 4627  ⟨cop 4633  ∪ cuni 4907   Se wse 5628   We wwe 5629  dom cdm 5675   ↾ cres 5677  Predcpred 6296   Fn wfn 6535  ‘cfv 6540  wrecscwrecs 8292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-2nd 7972  df-frecs 8262  df-wrecs 8293

This theorem is referenced by:  wfr1OLD  8323  wfr2OLD  8324