Theorem wfrlem10 7958

Description: Lemma for well-founded recursion. When 𝑧 is an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), then its predecessor class is equal to dom 𝐹. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem10.1	⊢ 𝑅 We 𝐴
wfrlem10.2	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfrlem10	⊢ ((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 wfrlem10

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem10.2	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2	1	wfrlem8 7956	. . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
3	2	biimpi 218	. 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4		predss 6149	. . . 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 487	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7		eldifn 4103	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8		eleq1w 2895	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹 ↔ 𝑧 ∈ dom 𝐹))
9	8	notbid 320	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
10	7, 9	syl5ibrcom 249	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
11	10	con2d 136	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
12	11	imp 409	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
13		ssel 3960	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) → 𝑧 ∈ dom 𝐹))
14	13	con3d 155	. . . . . . . 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	wfrdmcl 7957	. . . . . . 7 ⊢ (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
17	15, 16	impel 508	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
18		eldifi 4102	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
19		elpredg 6156	. . . . . . . 8 ⊢ ((𝑤 ∈ dom 𝐹 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
20	19	ancoms 461	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
21	18, 20	sylan 582	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
22	17, 21	mtbid 326	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
23	1	wfrdmss 7955	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
24	23	sseli 3962	. . . . . 6 ⊢ (𝑤 ∈ dom 𝐹 → 𝑤 ∈ 𝐴)
25		wfrlem10.1	. . . . . . . 8 ⊢ 𝑅 We 𝐴
26		weso 5540	. . . . . . . 8 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ 𝑅 Or 𝐴
28		solin 5492	. . . . . . 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 598	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧 ∨ 𝑤 = 𝑧 ∨ 𝑧𝑅𝑤))
31	12, 22, 30	ecase23d 1469	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
32		vex 3497	. . . . . 6 ⊢ 𝑤 ∈ V
33	32	elpred 6155	. . . . 5 ⊢ (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧)))
34	33	elv 3499	. . . 4 ⊢ (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹 ∧ 𝑤𝑅𝑧))
35	6, 31, 34	sylanbrc 585	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
36	5, 35	eqelssd 3987	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
37	3, 36	sylan9eqr 2878	1 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ w3o 1082   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290   class class class wbr 5058   Or wor 5467   We wwe 5507  dom cdm 5549  Predcpred 6141  wrecscwrecs 7940

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-so 5469  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357  df-wrecs 7941

This theorem is referenced by:  wfrlem15  7963