Theorem wfrlem10OLD 8317

Description: Lemma for well-ordered recursion. When 𝑧 is an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), then its predecessor class is equal to dom 𝐹. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem10OLD.1	⊢ 𝑅 We 𝐴
wfrlem10OLD.2	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfrlem10OLD	⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)

Distinct variable group:   𝑧,𝐴

Allowed substitution hints:   𝑅(𝑧)   𝐹(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem10OLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem10OLD.2	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2	1	wfrlem8OLD 8315	. . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
3	2	biimpi 215	. 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4		predss 6308	. . . 4 ⊢ Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
5	4	a1i 11	. . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹)
6		simpr 485	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7		eldifn 4127	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8		eleq1w 2816	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹 ↔ 𝑧 ∈ dom 𝐹))
9	8	notbid 317	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
10	7, 9	syl5ibrcom 246	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
11	10	con2d 134	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
12	11	imp 407	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
13		ssel 3975	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) → 𝑧 ∈ dom 𝐹))
14	13	con3d 152	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (¬ 𝑧 ∈ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
15	7, 14	syl5com 31	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
16	1	wfrdmclOLD 8316	. . . . . . 7 ⊢ (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
17	15, 16	impel 506	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
18		eldifi 4126	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
19		elpredg 6314	. . . . . . . 8 ⊢ ((𝑤 ∈ dom 𝐹 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
20	19	ancoms 459	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
21	18, 20	sylan 580	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
22	17, 21	mtbid 323	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
23	1	wfrdmssOLD 8314	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
24	23	sseli 3978	. . . . . 6 ⊢ (𝑤 ∈ dom 𝐹 → 𝑤 ∈ 𝐴)
25		wfrlem10OLD.1	. . . . . . . 8 ⊢ 𝑅 We 𝐴
26		weso 5667	. . . . . . . 8 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 𝑅 Or 𝐴
28		solin 5613	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
29	27, 28	mpan 688	. . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
30	24, 18, 29	syl2anr 597	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
31	12, 22, 30	ecase23d 1473	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
32		vex 3478	. . . . . 6 ⊢ 𝑤 ∈ V
33	32	elpred 6317	. . . . 5 ⊢ (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧)))
34	33	elv 3480	. . . 4 ⊢ (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧))
35	6, 31, 34	sylanbrc 583	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
36	5, 35	eqelssd 4003	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
37	3, 36	sylan9eqr 2794	1 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148   Or wor 5587   We wwe 5630  dom cdm 5676  Predcpred 6299  wrecscwrecs 8295

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-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

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-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-so 5589  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-2nd 7975  df-frecs 8265  df-wrecs 8296

This theorem is referenced by:  wfrlem15OLD  8322