Theorem wfrlem10OLD 8357

Description: Obsolete version as of 18-Nov-2024. 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. (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 8355	. . 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 6331	. . . 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 4142	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8		eleq1w 2822	. . . . . . . . 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 3989	. . . . . . . . 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 8356	. . . . . . 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 4141	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
19		elpredg 6337	. . . . . . . 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 580	. . . . . 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 8354	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
24	23	sseli 3991	. . . . . 6 ⊢ (𝑤 ∈ dom 𝐹 → 𝑤 ∈ 𝐴)
25		wfrlem10OLD.1	. . . . . . . 8 ⊢ 𝑅 We 𝐴
26		weso 5680	. . . . . . . 8 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 𝑅 Or 𝐴
28		solin 5623	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
29	27, 28	mpan 690	. . . . . 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 1472	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
32		vex 3482	. . . . . 6 ⊢ 𝑤 ∈ V
33	32	elpred 6340	. . . . 5 ⊢ (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧)))
34	33	elv 3483	. . . 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 4017	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
37	3, 36	sylan9eqr 2797	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 1085   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148   Or wor 5596   We wwe 5640  dom cdm 5689  Predcpred 6322  wrecscwrecs 8335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-so 5598  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336

This theorem is referenced by:  wfrlem15OLD  8362