Theorem fpwwe2lem8 10556

Description: Lemma for fpwwe2 10561. 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.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (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)
fpwwe2lem8.x	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem8.y	⊢ (𝜑 → 𝑌𝑊𝑆)
fpwwe2lem8.m	⊢ 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem8.n	⊢ 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem8.s	⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)

Assertion

Ref	Expression
fpwwe2lem8	⊢ (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)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem8

Step	Hyp	Ref	Expression
1		fpwwe2lem8.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋𝑊𝑅)
2		fpwwe2.1	. . . . . . . . . 10 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3	2	relopabiv 5766	. . . . . . . . 9 ⊢ Rel 𝑊
4	3	brrelex1i 5677	. . . . . . . 8 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V)
6		fpwwe2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	2, 6	fpwwe2lem2 10550	. . . . . . . . 9 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8	1, 7	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
9	8	simprld 778	. . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝑋)
10		fpwwe2lem8.m	. . . . . . . 8 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
11	10	oiiso 9446	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
12	5, 9, 11	syl2anc 591	. . . . . 6 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
13		isof1o 7271	. . . . . 6 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
14		f1ofo 6778	. . . . . 6 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → 𝑀:dom 𝑀–onto→𝑋)
15		forn 6746	. . . . . 6 ⊢ (𝑀:dom 𝑀–onto→𝑋 → ran 𝑀 = 𝑋)
16	12, 13, 14, 15	4syl 19	. . . . 5 ⊢ (𝜑 → ran 𝑀 = 𝑋)
17		fpwwe2.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
18		fpwwe2lem8.y	. . . . . . 7 ⊢ (𝜑 → 𝑌𝑊𝑆)
19		fpwwe2lem8.n	. . . . . . 7 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
20		fpwwe2lem8.s	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)
21	2, 6, 17, 1, 18, 10, 19, 20	fpwwe2lem7 10555	. . . . . 6 ⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))
22	21	rneqd 5887	. . . . 5 ⊢ (𝜑 → ran 𝑀 = ran (𝑁 ↾ dom 𝑀))
23	16, 22	eqtr3d 2778	. . . 4 ⊢ (𝜑 → 𝑋 = ran (𝑁 ↾ dom 𝑀))
24		df-ima 5634	. . . 4 ⊢ (𝑁 “ dom 𝑀) = ran (𝑁 ↾ dom 𝑀)
25	23, 24	eqtr4di 2794	. . 3 ⊢ (𝜑 → 𝑋 = (𝑁 “ dom 𝑀))
26		imassrn 6030	. . . 4 ⊢ (𝑁 “ dom 𝑀) ⊆ ran 𝑁
27	3	brrelex1i 5677	. . . . . . 7 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
28	18, 27	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
29	2, 6	fpwwe2lem2 10550	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
30	18, 29	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
31	30	simprld 778	. . . . . 6 ⊢ (𝜑 → 𝑆 We 𝑌)
32	19	oiiso 9446	. . . . . 6 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
33	28, 31, 32	syl2anc 591	. . . . 5 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
34		isof1o 7271	. . . . 5 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
35		f1ofo 6778	. . . . 5 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → 𝑁:dom 𝑁–onto→𝑌)
36		forn 6746	. . . . 5 ⊢ (𝑁:dom 𝑁–onto→𝑌 → ran 𝑁 = 𝑌)
37	33, 34, 35, 36	4syl 19	. . . 4 ⊢ (𝜑 → ran 𝑁 = 𝑌)
38	26, 37	sseqtrid 3959	. . 3 ⊢ (𝜑 → (𝑁 “ dom 𝑀) ⊆ 𝑌)
39	25, 38	eqsstrd 3951	. 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
40	8	simplrd 776	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
41		relxp 5639	. . . . 5 ⊢ Rel (𝑋 × 𝑋)
42		relss 5728	. . . . 5 ⊢ (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
43	40, 41, 42	mpisyl 21	. . . 4 ⊢ (𝜑 → Rel 𝑅)
44		relinxp 5760	. . . 4 ⊢ Rel (𝑆 ∩ (𝑌 × 𝑋))
45	43, 44	jctir 526	. . 3 ⊢ (𝜑 → (Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))))
46	40	ssbrd 5118	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝑥(𝑋 × 𝑋)𝑦))
47		brxp 5670	. . . . . . 7 ⊢ (𝑥(𝑋 × 𝑋)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
48	46, 47	imbitrdi 253	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
49		brinxp2 5699	. . . . . . 7 ⊢ (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦))
50		isocnv 7278	. . . . . . . . . . . . . 14 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
51	33, 50	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
52	51	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
53		isof1o 7271	. . . . . . . . . . . 12 ⊢ (◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) → ◡𝑁:𝑌–1-1-onto→dom 𝑁)
54		f1ofn 6772	. . . . . . . . . . . 12 ⊢ (◡𝑁:𝑌–1-1-onto→dom 𝑁 → ◡𝑁 Fn 𝑌)
55	52, 53, 54	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑁 Fn 𝑌)
56		simprll 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ 𝑌)
57		simprr 779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥𝑆𝑦)
58	39	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 ⊆ 𝑌)
59		simprlr 786	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ 𝑋)
60	58, 59	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ 𝑌)
61		isorel 7274	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑆𝑦 ↔ (◡𝑁‘𝑥) E (◡𝑁‘𝑦)))
62	52, 56, 60, 61	syl12anc 843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥𝑆𝑦 ↔ (◡𝑁‘𝑥) E (◡𝑁‘𝑦)))
63	57, 62	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) E (◡𝑁‘𝑦))
64		fvex 6844	. . . . . . . . . . . . . 14 ⊢ (◡𝑁‘𝑦) ∈ V
65	64	epeli 5523	. . . . . . . . . . . . 13 ⊢ ((◡𝑁‘𝑥) E (◡𝑁‘𝑦) ↔ (◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦))
66	63, 65	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦))
67	21	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑀 = (𝑁 ↾ dom 𝑀))
68	67	cnveqd 5820	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀 = ◡(𝑁 ↾ dom 𝑀))
69		fnfun 6589	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑁 Fn 𝑌 → Fun ◡𝑁)
70		funcnvres 6567	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ◡𝑁 → ◡(𝑁 ↾ dom 𝑀) = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
71	55, 69, 70	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡(𝑁 ↾ dom 𝑀) = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
72	68, 71	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀 = (◡𝑁 ↾ (𝑁 “ dom 𝑀)))
73	72	fveq1d 6833	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) = ((◡𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦))
74	25	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (𝑁 “ dom 𝑀))
75	59, 74	eleqtrd 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
76	75	fvresd 6851	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ((◡𝑁 ↾ (𝑁 “ dom 𝑀))‘𝑦) = (◡𝑁‘𝑦))
77	73, 76	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) = (◡𝑁‘𝑦))
78		isocnv 7278	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
79		isof1o 7271	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
80		f1of 6771	. . . . . . . . . . . . . . . 16 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
81	12, 78, 79, 80	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡𝑀:𝑋⟶dom 𝑀)
82	81	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → ◡𝑀:𝑋⟶dom 𝑀)
83	82, 59	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑀‘𝑦) ∈ dom 𝑀)
84	77, 83	eqeltrrd 2842	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑦) ∈ dom 𝑀)
85	10	oicl 9438	. . . . . . . . . . . . 13 ⊢ Ord dom 𝑀
86		ordtr1 6358	. . . . . . . . . . . . 13 ⊢ (Ord dom 𝑀 → (((◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦) ∧ (◡𝑁‘𝑦) ∈ dom 𝑀) → (◡𝑁‘𝑥) ∈ dom 𝑀))
87	85, 86	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((◡𝑁‘𝑥) ∈ (◡𝑁‘𝑦) ∧ (◡𝑁‘𝑦) ∈ dom 𝑀) → (◡𝑁‘𝑥) ∈ dom 𝑀)
88	66, 84, 87	syl2anc 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (◡𝑁‘𝑥) ∈ dom 𝑀)
89	55, 56, 88	elpreimad 7004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ (◡◡𝑁 “ dom 𝑀))
90		imacnvcnv 6161	. . . . . . . . . . 11 ⊢ (◡◡𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀)
91	74, 90	eqtr4di 2794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑋 = (◡◡𝑁 “ dom 𝑀))
92	89, 91	eleqtrrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → 𝑥 ∈ 𝑋)
93	92, 59	jca 517	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
94	93	ex 414	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
95	49, 94	biimtrid 244	. . . . . 6 ⊢ (𝜑 → (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
96	21	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑀 = (𝑁 ↾ dom 𝑀))
97	96	cnveqd 5820	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ◡𝑀 = ◡(𝑁 ↾ dom 𝑀))
98	97	fveq1d 6833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (◡𝑀‘𝑥) = (◡(𝑁 ↾ dom 𝑀)‘𝑥))
99	97	fveq1d 6833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (◡𝑀‘𝑦) = (◡(𝑁 ↾ dom 𝑀)‘𝑦))
100	98, 99	breq12d 5088	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((◡𝑀‘𝑥) E (◡𝑀‘𝑦) ↔ (◡(𝑁 ↾ dom 𝑀)‘𝑥) E (◡(𝑁 ↾ dom 𝑀)‘𝑦)))
101	12, 78	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
102		isorel 7274	. . . . . . . . . 10 ⊢ ((◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ (◡𝑀‘𝑥) E (◡𝑀‘𝑦)))
103	101, 102	sylan 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ (◡𝑀‘𝑥) E (◡𝑀‘𝑦)))
104		eqidd 2742	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 “ dom 𝑀) = (𝑁 “ dom 𝑀))
105		isores3 7283	. . . . . . . . . . . . 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)))
106	33, 20, 104, 105	syl3anc 1380	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)))
107		isocnv 7278	. . . . . . . . . . . 12 ⊢ ((𝑁 ↾ dom 𝑀) Isom E , 𝑆 (dom 𝑀, (𝑁 “ dom 𝑀)) → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
108	106, 107	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
109	108	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ◡(𝑁 ↾ dom 𝑀) Isom 𝑆, E ((𝑁 “ dom 𝑀), dom 𝑀))
110		simprl 777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
111	25	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 = (𝑁 “ dom 𝑀))
112	110, 111	eleqtrd 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (𝑁 “ dom 𝑀))
113		simprr 779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
114	113, 111	eleqtrd 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (𝑁 “ dom 𝑀))
115		isorel 7274	. . . . . . . . . 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)))
116	109, 112, 114, 115	syl12anc 843	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ (◡(𝑁 ↾ dom 𝑀)‘𝑥) E (◡(𝑁 ↾ dom 𝑀)‘𝑦)))
117	100, 103, 116	3bitr4d 313	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
118	39	sselda 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑌)
119	118	adantrr 724	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑌)
120	119, 113	jca 517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋))
121	120	biantrurd 538	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑆𝑦)))
122	121, 49	bitr4di 291	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑆𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
123	117, 122	bitrd 281	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
124	123	ex 414	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦)))
125	48, 95, 124	pm5.21ndd 381	. . . . 5 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ 𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦))
126		df-br 5076	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
127		df-br 5076	. . . . 5 ⊢ (𝑥(𝑆 ∩ (𝑌 × 𝑋))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋)))
128	125, 126, 127	3bitr3g 315	. . . 4 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑆 ∩ (𝑌 × 𝑋))))
129	128	eqrelrdv2 5741	. . 3 ⊢ (((Rel 𝑅 ∧ Rel (𝑆 ∩ (𝑌 × 𝑋))) ∧ 𝜑) → 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
130	45, 129	mpancom 695	. 2 ⊢ (𝜑 → 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))
131	39, 130	jca 517	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433  [wsbc 3725   ∩ cin 3884   ⊆ wss 3885  {csn 4558  ⟨cop 4564   class class class wbr 5075  {copab 5137   E cep 5520   We wwe 5573   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624  Rel wrel 5626  Ord word 6313  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  –onto→wfo 6487  –1-1-onto→wf1o 6488  ‘cfv 6489   Isom wiso 6490  (class class class)co 7360  OrdIsocoi 9418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-oi 9419

This theorem is referenced by:  fpwwe2lem9  10557