Theorem fpwwe2lem9 9460

Description: Lemma for fpwwe2 9465. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (The 𝑂 ⊆ 𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.)

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.x	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem9.y	⊢ (𝜑 → 𝑌𝑊𝑆)
fpwwe2lem9.m	⊢ 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n	⊢ 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem9.s	⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)

Assertion

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

Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝑀,𝑟,𝑢,𝑥,𝑦   𝑁,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)

Proof of Theorem fpwwe2lem9

Step	Hyp	Ref	Expression
1		fpwwe2lem9.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋𝑊𝑅)
2		fpwwe2.1	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3	2	relopabi 5245	. . . . . . . . . 10 ⊢ Rel 𝑊
4	3	brrelexi 5158	. . . . . . . . 9 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6		fpwwe2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
7	2, 6	fpwwe2lem2 9454	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8	1, 7	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
9	8	simprd 479	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
10	9	simpld 475	. . . . . . . 8 ⊢ (𝜑 → 𝑅 We 𝑋)
11		fpwwe2lem9.m	. . . . . . . . 9 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
12	11	oiiso 8442	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
13	5, 10, 12	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
14		isof1o 6573	. . . . . . 7 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
15	13, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:dom 𝑀–1-1-onto→𝑋)
16		f1ofo 6144	. . . . . 6 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → 𝑀:dom 𝑀–onto→𝑋)
17		forn 6118	. . . . . 6 ⊢ (𝑀:dom 𝑀–onto→𝑋 → ran 𝑀 = 𝑋)
18	15, 16, 17	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝑀 = 𝑋)
19		fpwwe2.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
20		fpwwe2lem9.y	. . . . . . 7 ⊢ (𝜑 → 𝑌𝑊𝑆)
21		fpwwe2lem9.n	. . . . . . 7 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
22		fpwwe2lem9.s	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)
23	2, 6, 19, 1, 20, 11, 21, 22	fpwwe2lem8 9459	. . . . . 6 ⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))
24	23	rneqd 5353	. . . . 5 ⊢ (𝜑 → ran 𝑀 = ran (𝑁 ↾ dom 𝑀))
25	18, 24	eqtr3d 2658	. . . 4 ⊢ (𝜑 → 𝑋 = ran (𝑁 ↾ dom 𝑀))
26		df-ima 5127	. . . 4 ⊢ (𝑁 “ dom 𝑀) = ran (𝑁 ↾ dom 𝑀)
27	25, 26	syl6eqr 2674	. . 3 ⊢ (𝜑 → 𝑋 = (𝑁 “ dom 𝑀))
28		imassrn 5477	. . . 4 ⊢ (𝑁 “ dom 𝑀) ⊆ ran 𝑁
29	3	brrelexi 5158	. . . . . . . 8 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
30	20, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
31	2, 6	fpwwe2lem2 9454	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
32	20, 31	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
33	32	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))
34	33	simpld 475	. . . . . . 7 ⊢ (𝜑 → 𝑆 We 𝑌)
35	21	oiiso 8442	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
36	30, 34, 35	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
37		isof1o 6573	. . . . . 6 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁:dom 𝑁–1-1-onto→𝑌)
39		f1ofo 6144	. . . . 5 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → 𝑁:dom 𝑁–onto→𝑌)
40		forn 6118	. . . . 5 ⊢ (𝑁:dom 𝑁–onto→𝑌 → ran 𝑁 = 𝑌)
41	38, 39, 40	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑁 = 𝑌)
42	28, 41	syl5sseq 3653	. . 3 ⊢ (𝜑 → (𝑁 “ dom 𝑀) ⊆ 𝑌)
43	27, 42	eqsstrd 3639	. 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
44	8	simpld 475	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)))
45	44	simprd 479	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
46		relxp 5227	. . . . 5 ⊢ Rel (𝑋 × 𝑋)
47		relss 5206	. . . . 5 ⊢ (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
48	45, 46, 47	mpisyl 21	. . . 4 ⊢ (𝜑 → Rel 𝑅)
49		inss2 3834	. . . . 5 ⊢ (𝑆 ∩ (𝑌 × 𝑋)) ⊆ (𝑌 × 𝑋)
50		relxp 5227	. . . . 5 ⊢ Rel (𝑌 × 𝑋)
51		relss 5206	. . . . 5 ⊢ ((𝑆 ∩ (𝑌 × 𝑋)) ⊆ (𝑌 × 𝑋) → (Rel (𝑌 × 𝑋) → Rel (𝑆 ∩ (𝑌 × 𝑋))))
52	49, 50, 51	mp2 9	. . . 4 ⊢ Rel (𝑆 ∩ (𝑌 × 𝑋))
53	48, 52	jctir 561	. . 3 ⊢ (𝜑 → (Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))))
54	45	ssbrd 4696	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑥(𝑋 × 𝑋)𝑦))
55		brxp 5147	. . . . . . 7 ⊢ (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
56	54, 55	syl6ib 241	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
57		brinxp2 5180	. . . . . . . 8 ⊢ (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥𝑆𝑦))
58		df-3an 1039	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥𝑆𝑦) ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦))
59	57, 58	bitri 264	. . . . . . 7 ⊢ (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦))
60		simprll 802	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ 𝑌)
61		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥𝑆𝑦)
62		isocnv 6580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
63	36, 62	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
64	63	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
65	43	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 ⊆ 𝑌)
66		simprlr 803	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ 𝑋)
67	65, 66	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ 𝑌)
68		isorel 6576	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑆𝑦 ↔ (◡𝑁‘𝑥) E (◡𝑁‘𝑦)))
69	64, 60, 67, 68	syl12anc 1324	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥𝑆𝑦 ↔ (◡𝑁‘𝑥) E (◡𝑁‘𝑦)))
70	61, 69	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) E (◡𝑁‘𝑦))
71		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (◡𝑁‘𝑦) ∈ V
72	71	epelc 5031	. . . . . . . . . . . . 13 ⊢ ((◡𝑁‘𝑥) E (◡𝑁‘𝑦) ↔ (◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦))
73	70, 72	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦))
74	23	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀 = (𝑁 ↾ dom 𝑀))
75	74	cnveqd 5298	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀 = ◡(𝑁 ↾ dom 𝑀))
76		isof1o 6573	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) → ◡𝑁:𝑌–1-1-onto→dom 𝑁)
77		f1ofn 6138	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑁:𝑌–1-1-onto→dom 𝑁 → ◡𝑁 Fn 𝑌)
78	64, 76, 77	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑁 Fn 𝑌)
79		fnfun 5988	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑁 Fn 𝑌 → Fun ◡𝑁)
80		funcnvres 5967	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ◡𝑁 → ◡(𝑁 ↾ dom 𝑀) = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
81	78, 79, 80	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡(𝑁 ↾ dom 𝑀) = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
82	75, 81	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀 = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
83	82	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) = ((◡𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦))
84	27	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (𝑁 “ dom 𝑀))
85	66, 84	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
86		fvres 6207	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝑁 “ dom 𝑀) → ((◡𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦) = (◡𝑁‘𝑦))
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ((◡𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦) = (◡𝑁‘𝑦))
88	83, 87	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) = (◡𝑁‘𝑦))
89		isocnv 6580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
90	13, 89	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
91		isof1o 6573	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
92		f1of 6137	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
93	90, 91, 92	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡𝑀:𝑋⟶dom 𝑀)
94	93	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀:𝑋⟶dom 𝑀)
95	94, 66	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) ∈ dom 𝑀)
96	88, 95	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑦) ∈ dom 𝑀)
97	11	oicl 8434	. . . . . . . . . . . . 13 ⊢ Ord dom 𝑀
98		ordtr1 5767	. . . . . . . . . . . . 13 ⊢ (Ord dom 𝑀 → (((◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦) ∧ (◡𝑁‘𝑦) ∈ dom 𝑀) → (◡𝑁‘𝑥) ∈ dom 𝑀))
99	97, 98	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦) ∧ (◡𝑁‘𝑦) ∈ dom 𝑀) → (◡𝑁‘𝑥) ∈ dom 𝑀)
100	73, 96, 99	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) ∈ dom 𝑀)
101		elpreima 6337	. . . . . . . . . . . 12 ⊢ (◡𝑁 Fn 𝑌 → (𝑥 ∈ (◡◡𝑁 “ dom 𝑀) ↔ (𝑥 ∈ 𝑌 ∧ (◡𝑁‘𝑥) ∈ dom 𝑀)))
102	78, 101	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥 ∈ (◡◡𝑁 “ dom 𝑀) ↔ (𝑥 ∈ 𝑌 ∧ (◡𝑁‘𝑥) ∈ dom 𝑀)))
103	60, 100, 102	mpbir2and 957	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ (◡◡𝑁 “ dom 𝑀))
104		imacnvcnv 5599	. . . . . . . . . . 11 ⊢ (◡◡𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀)
105	84, 104	syl6eqr 2674	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (◡◡𝑁 “ dom 𝑀))
106	103, 105	eleqtrrd 2704	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ 𝑋)
107	106, 66	jca 554	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
108	107	ex 450	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
109	59, 108	syl5bi 232	. . . . . 6 ⊢ (𝜑 → (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
110	23	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑀 = (𝑁 ↾ dom 𝑀))
111	110	cnveqd 5298	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ◡𝑀 = ◡(𝑁 ↾ dom 𝑀))
112	111	fveq1d 6193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (◡𝑀‘𝑥) = (◡(𝑁 ↾ dom 𝑀)‘𝑥))
113	111	fveq1d 6193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (◡𝑀‘𝑦) = (◡(𝑁 ↾ dom 𝑀)‘𝑦))
114	112, 113	breq12d 4666	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝑀‘𝑥) E (◡𝑀‘𝑦) ↔ (◡(𝑁 ↾ dom 𝑀)‘𝑥) E (◡(𝑁 ↾ dom 𝑀)‘𝑦)))
115		isorel 6576	. . . . . . . . . 10 ⊢ ((◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ (◡𝑀‘𝑥) E (◡𝑀‘𝑦)))
116	90, 115	sylan 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ (◡𝑀‘𝑥) E (◡𝑀‘𝑦)))
117		eqidd 2623	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀))
118		isores3 6585	. . . . . . . . . . . . 13 ⊢ ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ dom 𝑀 ⊆ dom 𝑁 ∧ (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀)) → (𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)))
119	36, 22, 117, 118	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)))
120		isocnv 6580	. . . . . . . . . . . 12 ⊢ ((𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)) → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
121	119, 120	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
122	121	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
123		simprl 794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
124	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 = (𝑁 “ dom 𝑀))
125	123, 124	eleqtrd 2703	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (𝑁 “ dom 𝑀))
126		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
127	126, 124	eleqtrd 2703	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
128		isorel 6576	. . . . . . . . . 10 ⊢ ((◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀) ∧ (𝑥 ∈ (𝑁 “ dom 𝑀) ∧ 𝑦 ∈ (𝑁 “ dom 𝑀))) → (𝑥𝑆𝑦 ↔ (◡(𝑁 ↾ dom 𝑀)‘𝑥) E (◡(𝑁 ↾ dom 𝑀)‘𝑦)))
129	122, 125, 127, 128	syl12anc 1324	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ (◡(𝑁 ↾ dom 𝑀)‘𝑥) E (◡(𝑁 ↾ dom 𝑀)‘𝑦)))
130	114, 116, 129	3bitr4d 300	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
131	43	sselda 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑌)
132	131	adantrr 753	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑌)
133	132, 126	jca 554	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋))
134	133	biantrurd 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)))
135	134, 59	syl6bbr 278	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
136	130, 135	bitrd 268	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
137	136	ex 450	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦)))
138	56, 109, 137	pm5.21ndd 369	. . . . 5 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
139		df-br 4654	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
140		df-br 4654	. . . . 5 ⊢ (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋)))
141	138, 139, 140	3bitr3g 302	. . . 4 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋))))
142	141	eqrelrdv2 5219	. . 3 ⊢ (((Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))) ∧ 𝜑) → 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
143	53, 142	mpancom 703	. 2 ⊢ (𝜑 → 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
144	43, 143	jca 554	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ⟨cop 4183   class class class wbr 4653  {copab 4712   E cep 5028   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Rel wrel 5119  Ord word 5722  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  fpwwe2lem10  9461