Theorem fpwwe2lem7 10590

Description: Lemma for fpwwe2 10596. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (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
fpwwe2lem7	⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))

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 fpwwe2lem7

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwwe2lem8.m	. . . 4 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
2	1	oif 9483	. . 3 ⊢ 𝑀:dom 𝑀⟶𝑋
3		ffn 6688	. . 3 ⊢ (𝑀:dom 𝑀⟶𝑋 → 𝑀 Fn dom 𝑀)
4	2, 3	mp1i 13	. 2 ⊢ (𝜑 → 𝑀 Fn dom 𝑀)
5		fpwwe2lem8.n	. . . . 5 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
6	5	oif 9483	. . . 4 ⊢ 𝑁:dom 𝑁⟶𝑌
7		ffn 6688	. . . 4 ⊢ (𝑁:dom 𝑁⟶𝑌 → 𝑁 Fn dom 𝑁)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → 𝑁 Fn dom 𝑁)
9		fpwwe2lem8.s	. . 3 ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)
10	8, 9	fnssresd 6642	. 2 ⊢ (𝜑 → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
11	1	oicl 9482	. . . . . 6 ⊢ Ord dom 𝑀
12		ordelon 6356	. . . . . 6 ⊢ ((Ord dom 𝑀 ∧ 𝑤 ∈ dom 𝑀) → 𝑤 ∈ On)
13	11, 12	mpan 690	. . . . 5 ⊢ (𝑤 ∈ dom 𝑀 → 𝑤 ∈ On)
14		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ dom 𝑀 ↔ 𝑦 ∈ dom 𝑀))
15		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑀‘𝑤) = (𝑀‘𝑦))
16		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑁‘𝑤) = (𝑁‘𝑦))
17	15, 16	eqeq12d 2745	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑀‘𝑤) = (𝑁‘𝑤) ↔ (𝑀‘𝑦) = (𝑁‘𝑦)))
18	14, 17	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤)) ↔ (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))))
19	18	imbi2d 340	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤))) ↔ (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)))))
20		r19.21v 3158	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))) ↔ (𝜑 → ∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))))
21	11	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Ord dom 𝑀)
22		ordelss 6348	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord dom 𝑀 ∧ 𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
23	21, 22	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
24	23	sselda 3946	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ 𝑦 ∈ 𝑤) → 𝑦 ∈ dom 𝑀)
25		pm2.27 42	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑀 → ((𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → (𝑀‘𝑦) = (𝑁‘𝑦)))
26	24, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ 𝑦 ∈ 𝑤) → ((𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → (𝑀‘𝑦) = (𝑁‘𝑦)))
27	26	ralimdva 3145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → ∀𝑦 ∈ 𝑤 (𝑀‘𝑦) = (𝑁‘𝑦)))
28		fnssres 6641	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 Fn dom 𝑀 ∧ 𝑤 ⊆ dom 𝑀) → (𝑀 ↾ 𝑤) Fn 𝑤)
29	4, 23, 28	syl2an2r 685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑀 ↾ 𝑤) Fn 𝑤)
30	9	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → dom 𝑀 ⊆ dom 𝑁)
31	23, 30	sstrd 3957	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑁)
32		fnssres 6641	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 Fn dom 𝑁 ∧ 𝑤 ⊆ dom 𝑁) → (𝑁 ↾ 𝑤) Fn 𝑤)
33	8, 31, 32	syl2an2r 685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑁 ↾ 𝑤) Fn 𝑤)
34		eqfnfv 7003	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ↾ 𝑤) Fn 𝑤 ∧ (𝑁 ↾ 𝑤) Fn 𝑤) → ((𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤) ↔ ∀𝑦 ∈ 𝑤 ((𝑀 ↾ 𝑤)‘𝑦) = ((𝑁 ↾ 𝑤)‘𝑦)))
35	29, 33, 34	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → ((𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤) ↔ ∀𝑦 ∈ 𝑤 ((𝑀 ↾ 𝑤)‘𝑦) = ((𝑁 ↾ 𝑤)‘𝑦)))
36		fvres 6877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → ((𝑀 ↾ 𝑤)‘𝑦) = (𝑀‘𝑦))
37		fvres 6877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → ((𝑁 ↾ 𝑤)‘𝑦) = (𝑁‘𝑦))
38	36, 37	eqeq12d 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑤 → (((𝑀 ↾ 𝑤)‘𝑦) = ((𝑁 ↾ 𝑤)‘𝑦) ↔ (𝑀‘𝑦) = (𝑁‘𝑦)))
39	38	ralbiia 3073	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑤 ((𝑀 ↾ 𝑤)‘𝑦) = ((𝑁 ↾ 𝑤)‘𝑦) ↔ ∀𝑦 ∈ 𝑤 (𝑀‘𝑦) = (𝑁‘𝑦))
40	35, 39	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → ((𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤) ↔ ∀𝑦 ∈ 𝑤 (𝑀‘𝑦) = (𝑁‘𝑦)))
41		fpwwe2.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
42		fpwwe2.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
43	42	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝐴 ∈ 𝑉)
44		simpll 766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝜑)
45		fpwwe2.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
46	44, 45	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
47		fpwwe2lem8.x	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋𝑊𝑅)
48	47	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝑋𝑊𝑅)
49		fpwwe2lem8.y	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌𝑊𝑆)
50	49	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝑌𝑊𝑆)
51		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝑤 ∈ dom 𝑀)
52	9	sselda 3946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → 𝑤 ∈ dom 𝑁)
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → 𝑤 ∈ dom 𝑁)
54		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤))
55	41, 43, 46, 48, 50, 1, 5, 51, 53, 54	fpwwe2lem6 10589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ 𝑦𝑅(𝑀‘𝑤)) → (𝑦𝑆(𝑁‘𝑤) ∧ (𝑧𝑅(𝑀‘𝑤) → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))))
56	55	simpld 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ 𝑦𝑅(𝑀‘𝑤)) → 𝑦𝑆(𝑁‘𝑤))
57	54	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑁 ↾ 𝑤) = (𝑀 ↾ 𝑤))
58	41, 43, 46, 50, 48, 5, 1, 53, 51, 57	fpwwe2lem6 10589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ 𝑦𝑆(𝑁‘𝑤)) → (𝑦𝑅(𝑀‘𝑤) ∧ (𝑧𝑆(𝑁‘𝑤) → (𝑦𝑆𝑧 ↔ 𝑦𝑅𝑧))))
59	58	simpld 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ 𝑦𝑆(𝑁‘𝑤)) → 𝑦𝑅(𝑀‘𝑤))
60	56, 59	impbida 800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑦𝑅(𝑀‘𝑤) ↔ 𝑦𝑆(𝑁‘𝑤)))
61		fvex 6871	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀‘𝑤) ∈ V
62		vex 3451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
63	62	eliniseg 6065	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀‘𝑤) ∈ V → (𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ↔ 𝑦𝑅(𝑀‘𝑤)))
64	61, 63	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ↔ 𝑦𝑅(𝑀‘𝑤))
65		fvex 6871	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁‘𝑤) ∈ V
66	62	eliniseg 6065	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁‘𝑤) ∈ V → (𝑦 ∈ (◡𝑆 “ {(𝑁‘𝑤)}) ↔ 𝑦𝑆(𝑁‘𝑤)))
67	65, 66	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (◡𝑆 “ {(𝑁‘𝑤)}) ↔ 𝑦𝑆(𝑁‘𝑤))
68	60, 64, 67	3bitr4g 314	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ↔ 𝑦 ∈ (◡𝑆 “ {(𝑁‘𝑤)})))
69	68	eqrdv 2727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (◡𝑅 “ {(𝑀‘𝑤)}) = (◡𝑆 “ {(𝑁‘𝑤)}))
70		relinxp 5777	. . . . . . . . . . . . . . . . . . 19 ⊢ Rel (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))
71		relinxp 5777	. . . . . . . . . . . . . . . . . . 19 ⊢ Rel (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))
72		vex 3451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑧 ∈ V
73	72	eliniseg 6065	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑀‘𝑤) ∈ V → (𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ↔ 𝑧𝑅(𝑀‘𝑤)))
74	63, 73	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀‘𝑤) ∈ V → ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ↔ (𝑦𝑅(𝑀‘𝑤) ∧ 𝑧𝑅(𝑀‘𝑤))))
75	61, 74	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ↔ (𝑦𝑅(𝑀‘𝑤) ∧ 𝑧𝑅(𝑀‘𝑤)))
76	55	simprd 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ 𝑦𝑅(𝑀‘𝑤)) → (𝑧𝑅(𝑀‘𝑤) → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧)))
77	76	impr 454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ (𝑦𝑅(𝑀‘𝑤) ∧ 𝑧𝑅(𝑀‘𝑤))) → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
78	75, 77	sylan2b 594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) ∧ (𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)}))) → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
79	78	pm5.32da 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑆𝑧)))
80		df-br 5108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))))
81		brinxp2 5716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))𝑧 ↔ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑅𝑧))
82	80, 81	bitr3i 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) ↔ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑅𝑧))
83		df-br 5108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))))
84		brinxp2 5716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))𝑧 ↔ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑆𝑧))
85	83, 84	bitr3i 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) ↔ ((𝑦 ∈ (◡𝑅 “ {(𝑀‘𝑤)}) ∧ 𝑧 ∈ (◡𝑅 “ {(𝑀‘𝑤)})) ∧ 𝑦𝑆𝑧))
86	79, 82, 85	3bitr4g 314	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))))
87	70, 71, 86	eqrelrdv 5755	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) = (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))))
88	69	sqxpeqd 5670	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})) = ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)})))
89	88	ineq2d 4183	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑆 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) = (𝑆 ∩ ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)}))))
90	87, 89	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)}))) = (𝑆 ∩ ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)}))))
91	69, 90	oveq12d 7405	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → ((◡𝑅 “ {(𝑀‘𝑤)})𝐹(𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))) = ((◡𝑆 “ {(𝑁‘𝑤)})𝐹(𝑆 ∩ ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)})))))
92	2	ffvelcdmi 7055	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) ∈ 𝑋)
93	92	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑀‘𝑤) ∈ 𝑋)
94	93	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑀‘𝑤) ∈ 𝑋)
95	41, 42, 47	fpwwe2lem3 10586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑀‘𝑤) ∈ 𝑋) → ((◡𝑅 “ {(𝑀‘𝑤)})𝐹(𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))) = (𝑀‘𝑤))
96	44, 94, 95	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → ((◡𝑅 “ {(𝑀‘𝑤)})𝐹(𝑅 ∩ ((◡𝑅 “ {(𝑀‘𝑤)}) × (◡𝑅 “ {(𝑀‘𝑤)})))) = (𝑀‘𝑤))
97	6	ffvelcdmi 7055	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ dom 𝑁 → (𝑁‘𝑤) ∈ 𝑌)
98	52, 97	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑁‘𝑤) ∈ 𝑌)
99	98	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑁‘𝑤) ∈ 𝑌)
100	41, 42, 49	fpwwe2lem3 10586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑁‘𝑤) ∈ 𝑌) → ((◡𝑆 “ {(𝑁‘𝑤)})𝐹(𝑆 ∩ ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)})))) = (𝑁‘𝑤))
101	44, 99, 100	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → ((◡𝑆 “ {(𝑁‘𝑤)})𝐹(𝑆 ∩ ((◡𝑆 “ {(𝑁‘𝑤)}) × (◡𝑆 “ {(𝑁‘𝑤)})))) = (𝑁‘𝑤))
102	91, 96, 101	3eqtr3d 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ dom 𝑀) ∧ (𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤)) → (𝑀‘𝑤) = (𝑁‘𝑤))
103	102	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → ((𝑀 ↾ 𝑤) = (𝑁 ↾ 𝑤) → (𝑀‘𝑤) = (𝑁‘𝑤)))
104	40, 103	sylbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (∀𝑦 ∈ 𝑤 (𝑀‘𝑦) = (𝑁‘𝑦) → (𝑀‘𝑤) = (𝑁‘𝑤)))
105	27, 104	syld 47	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → (𝑀‘𝑤) = (𝑁‘𝑤)))
106	105	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ dom 𝑀 → (∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → (𝑀‘𝑤) = (𝑁‘𝑤))))
107	106	com23 86	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦)) → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤))))
108	107	a2i 14	. . . . . . . . 9 ⊢ ((𝜑 → ∀𝑦 ∈ 𝑤 (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤))))
109	20, 108	sylbi 217	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤))))
110	109	a1i 11	. . . . . . 7 ⊢ (𝑤 ∈ On → (∀𝑦 ∈ 𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀‘𝑦) = (𝑁‘𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤)))))
111	19, 110	tfis2 7833	. . . . . 6 ⊢ (𝑤 ∈ On → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤))))
112	111	com3l 89	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑤 ∈ On → (𝑀‘𝑤) = (𝑁‘𝑤))))
113	13, 112	mpdi 45	. . . 4 ⊢ (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀‘𝑤) = (𝑁‘𝑤)))
114	113	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑀‘𝑤) = (𝑁‘𝑤))
115		fvres 6877	. . . 4 ⊢ (𝑤 ∈ dom 𝑀 → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁‘𝑤))
116	115	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁‘𝑤))
117	114, 116	eqtr4d 2767	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ dom 𝑀) → (𝑀‘𝑤) = ((𝑁 ↾ dom 𝑀)‘𝑤))
118	4, 10, 117	eqfnfvd 7006	1 ⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  [wsbc 3753   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ⟨cop 4595   class class class wbr 5107  {copab 5169   We wwe 5590   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   ↾ cres 5640   “ cima 5641  Ord word 6331  Oncon0 6332   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  OrdIsocoi 9462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-oi 9463

This theorem is referenced by:  fpwwe2lem8  10591