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Theorem fpwwe2lem9 10680
Description: Lemma for fpwwe2 10684. 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⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴𝑉)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem9.4 (𝜑𝑋𝑊𝑅)
fpwwe2lem9.6 (𝜑𝑌𝑊𝑆)
Assertion
Ref Expression
fpwwe2lem9 (𝜑 → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (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
StepHypRef Expression
1 eqid 2736 . . . 4 OrdIso(𝑅, 𝑋) = OrdIso(𝑅, 𝑋)
21oicl 9570 . . 3 Ord dom OrdIso(𝑅, 𝑋)
3 eqid 2736 . . . 4 OrdIso(𝑆, 𝑌) = OrdIso(𝑆, 𝑌)
43oicl 9570 . . 3 Ord dom OrdIso(𝑆, 𝑌)
5 ordtri2or2 6482 . . 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)))
62, 4, 5mp2an 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⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
8 fpwwe2.2 . . . . . 6 (𝜑𝐴𝑉)
98adantr 480 . . . . 5 ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝐴𝑉)
10 fpwwe2.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
1110adantlr 715 . . . . 5 (((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
12 fpwwe2lem9.4 . . . . . 6 (𝜑𝑋𝑊𝑅)
1312adantr 480 . . . . 5 ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑋𝑊𝑅)
14 fpwwe2lem9.6 . . . . . 6 (𝜑𝑌𝑊𝑆)
1514adantr 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))
177, 9, 11, 13, 15, 1, 3, 16fpwwe2lem8 10679 . . . 4 ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
1817ex 412 . . 3 (𝜑 → (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))))
198adantr 480 . . . . 5 ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝐴𝑉)
2010adantlr 715 . . . . 5 (((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
2114adantr 480 . . . . 5 ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑌𝑊𝑆)
2212adantr 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))
247, 19, 20, 21, 22, 3, 1, 23fpwwe2lem8 10679 . . . 4 ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))
2524ex 412 . . 3 (𝜑 → (dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋) → (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
2618, 25orim12d 966 . 2 (𝜑 → ((dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) ∨ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))))
276, 26mpi 20 1 (𝜑 → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wral 3060  [wsbc 3787  cin 3949  wss 3950  {csn 4625   class class class wbr 5142  {copab 5204   We wwe 5635   × cxp 5682  ccnv 5683  dom cdm 5684  cima 5687  Ord word 6382  (class class class)co 7432  OrdIsocoi 9550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-oi 9551
This theorem is referenced by:  fpwwe2lem10  10681  fpwwe2lem11  10682  fpwwe2  10684
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