Theorem fpwwe2lem9 10553

Description: Lemma for fpwwe2 10557. 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 2739	. . . 4 ⊢ OrdIso(𝑅, 𝑋) = OrdIso(𝑅, 𝑋)
2	1	oicl 9434	. . 3 ⊢ Ord dom OrdIso(𝑅, 𝑋)
3		eqid 2739	. . . 4 ⊢ OrdIso(𝑆, 𝑌) = OrdIso(𝑆, 𝑌)
4	3	oicl 9434	. . 3 ⊢ Ord dom OrdIso(𝑆, 𝑌)
5		ordtri2or2 6411	. . 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 698	. 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 481	. . . . 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 721	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑋𝑊𝑅)
14		fpwwe2lem9.6	. . . . . 6 ⊢ (𝜑 → 𝑌𝑊𝑆)
15	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → 𝑌𝑊𝑆)
16		simpr 485	. . . . 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 10552	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌)) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
18	17	ex 413	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑅, 𝑋) ⊆ dom OrdIso(𝑆, 𝑌) → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))))
19	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝐴 ∈ 𝑉)
20	10	adantlr 721	. . . . 5 ⊢ (((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
21	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑌𝑊𝑆)
22	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → 𝑋𝑊𝑅)
23		simpr 485	. . . . 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 10552	. . . 4 ⊢ ((𝜑 ∧ dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋)) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))
25	24	ex 413	. . 3 ⊢ (𝜑 → (dom OrdIso(𝑆, 𝑌) ⊆ dom OrdIso(𝑅, 𝑋) → (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
26	18, 25	orim12d 972	. 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 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  [wsbc 3723   ∩ cin 3882   ⊆ wss 3883  {csn 4555   class class class wbr 5072  {copab 5134   We wwe 5570   × cxp 5616  ^◡ccnv 5617  dom cdm 5618   “ cima 5621  Ord word 6309  (class class class)co 7356  OrdIsocoi 9414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-oi 9415

This theorem is referenced by:  fpwwe2lem10  10554  fpwwe2lem11  10555  fpwwe2  10557