Theorem fpwwe2lem9 10731

Description: Lemma for fpwwe2 10735. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem9.4	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem9.6	⊢ (𝜑 → 𝑌𝑊𝑆)

Assertion

Ref	Expression
fpwwe2lem9	⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))

Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑆,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦, 𝑢)   𝑉(𝑥, 𝑦, 𝑢, 𝑟)

Proof of Theorem fpwwe2lem9

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ OrdIso(𝑅, 𝑋) = OrdIso(𝑅, 𝑋)
2	1	oicl 9530	. . 3 ⊢ Ord dom OrdIso(𝑅, 𝑋)
3		eqid 2760	. . . 4 ⊢ OrdIso(𝑆, 𝑌) = OrdIso(𝑆, 𝑌)
4	3	oicl 9530	. . 3 ⊢ Ord dom OrdIso(𝑆, 𝑌)
5		ordtri2or2 6467	. . 3 ⊢ ((Ord dom OrdIso(𝑅, 𝑋) ∧ Ord dom OrdIso(𝑆, 𝑌)) → (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) ∨ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)))
6	2, 4, 5	mp2an 705	. 2 ⊢ (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) ∨ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋))
7		fpwwe2.1	. . . . 5 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
8		fpwwe2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝐴 ∈ 𝑉)
10		fpwwe2.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
11	10	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
12		fpwwe2lem9.4	. . . . . 6 ⊢ (𝜑 → 𝑋𝑊𝑅)
13	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑋𝑊𝑅)
14		fpwwe2lem9.6	. . . . . 6 ⊢ (𝜑 → 𝑌𝑊𝑆)
15	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑌𝑊𝑆)
16		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌))
17	7, 9, 11, 13, 15, 1, 3, 16	fpwwe2lem8 10730	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
18	17	ex 418	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))))
19	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝐴 ∈ 𝑉)
20	10	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
21	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑌𝑊𝑆)
22	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑋𝑊𝑅)
23		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋))
24	7, 19, 20, 21, 22, 3, 1, 23	fpwwe2lem8 10730	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))
25	24	ex 418	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
26	18, 25	orim12d 979	. 2 ⊢ (𝜑 → ((dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) ∨ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))))
27	6, 26	mpi 21	1 ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  [wsbc 3738   ⊆ wss 3896   ∩ cin 4088  {csn 4584   class class class wbr 5102  {copab 5166   We wwe 5603   × cxp 5649  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  Ord word 6364  (class class class)co 7424  OrdIsocoi 9510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-oi 9511

This theorem is used by:  fpwwe2lem10  10732  fpwwe2lem11  10733  fpwwe2  10735