Theorem fpwwe2lem9 10658

Description: Lemma for fpwwe2 10662. 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 2736	. . . 4 ⊢ OrdIso(𝑅, 𝑋) = OrdIso(𝑅, 𝑋)
2	1	oicl 9548	. . 3 ⊢ Ord dom OrdIso(𝑅, 𝑋)
3		eqid 2736	. . . 4 ⊢ OrdIso(𝑆, 𝑌) = OrdIso(𝑆, 𝑌)
4	3	oicl 9548	. . 3 ⊢ Ord dom OrdIso(𝑆, 𝑌)
5		ordtri2or2 6458	. . 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 692	. 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 480	. . . . 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 715	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑋𝑊𝑅)
14		fpwwe2lem9.6	. . . . . 6 ⊢ (𝜑 → 𝑌𝑊𝑆)
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑌𝑊𝑆)
16		simpr 484	. . . . 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 10657	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
18	17	ex 412	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))))
19	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝐴 ∈ 𝑉)
20	10	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
21	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑌𝑊𝑆)
22	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑋𝑊𝑅)
23		simpr 484	. . . . 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 10657	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))
25	24	ex 412	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
26	18, 25	orim12d 966	. 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 20	1 ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  [wsbc 3770   ∩ cin 3930   ⊆ wss 3931  {csn 4606   class class class wbr 5124  {copab 5186   We wwe 5610   × cxp 5657  ^◡ccnv 5658  dom cdm 5659   “ cima 5662  Ord word 6356  (class class class)co 7410  OrdIsocoi 9528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-oi 9529

This theorem is referenced by:  fpwwe2lem10  10659  fpwwe2lem11  10660  fpwwe2  10662