Theorem wfrlem10OLD 8374

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 8372	. . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
3	2	biimpi 216	. 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4		predss 6340	. . . 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 484	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7		eldifn 4155	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8		eleq1w 2827	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹 ↔ 𝑧 ∈ dom 𝐹))
9	8	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
10	7, 9	syl5ibrcom 247	. . . . . . 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 406	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
13		ssel 4002	. . . . . . . . 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 8373	. . . . . . 7 ⊢ (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
17	15, 16	impel 505	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
18		eldifi 4154	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
19		elpredg 6346	. . . . . . . 8 ⊢ ((𝑤 ∈ dom 𝐹 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
20	19	ancoms 458	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
21	18, 20	sylan 579	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
22	17, 21	mtbid 324	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
23	1	wfrdmssOLD 8371	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
24	23	sseli 4004	. . . . . 6 ⊢ (𝑤 ∈ dom 𝐹 → 𝑤 ∈ 𝐴)
25		wfrlem10OLD.1	. . . . . . . 8 ⊢ 𝑅 We 𝐴
26		weso 5691	. . . . . . . 8 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 𝑅 Or 𝐴
28		solin 5634	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
29	27, 28	mpan 689	. . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
30	24, 18, 29	syl2anr 596	. . . . 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 3492	. . . . . 6 ⊢ 𝑤 ∈ V
33	32	elpred 6349	. . . . 5 ⊢ (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧)))
34	33	elv 3493	. . . 4 ⊢ (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧))
35	6, 31, 34	sylanbrc 582	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
36	5, 35	eqelssd 4030	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
37	3, 36	sylan9eqr 2802	1 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   Or wor 5606   We wwe 5651  dom cdm 5700  Predcpred 6331  wrecscwrecs 8352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-so 5608  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353

This theorem is referenced by:  wfrlem15OLD  8379