Theorem fpwwe2lem9 10636

Description: Lemma for fpwwe2 10640. 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 2726	. . . 4 ⊢ OrdIso(𝑅, 𝑋) = OrdIso(𝑅, 𝑋)
2	1	oicl 9526	. . 3 ⊢ Ord dom OrdIso(𝑅, 𝑋)
3		eqid 2726	. . . 4 ⊢ OrdIso(𝑆, 𝑌) = OrdIso(𝑆, 𝑌)
4	3	oicl 9526	. . 3 ⊢ Ord dom OrdIso(𝑆, 𝑌)
5		ordtri2or2 6457	. . 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 689	. 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 712	. . . . 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 10635	. . . 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 712	. . . . 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 10635	. . . 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 961	. 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 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  [wsbc 3772   ∩ cin 3942   ⊆ wss 3943  {csn 4623   class class class wbr 5141  {copab 5203   We wwe 5623   × cxp 5667  ^◡ccnv 5668  dom cdm 5669   “ cima 5672  Ord word 6357  (class class class)co 7405  OrdIsocoi 9506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-oi 9507

This theorem is referenced by:  fpwwe2lem10  10637  fpwwe2lem11  10638  fpwwe2  10640