MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe2lem8 Structured version   Visualization version   GIF version

Theorem fpwwe2lem8 9712
Description: Lemma for fpwwe2 9718. 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.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem9.x (𝜑𝑋𝑊𝑅)
fpwwe2lem9.y (𝜑𝑌𝑊𝑆)
fpwwe2lem9.m 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem9.s (𝜑 → dom 𝑀 ⊆ dom 𝑁)
Assertion
Ref Expression
fpwwe2lem8 (𝜑𝑀 = (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)

Proof of Theorem fpwwe2lem8
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2lem9.m . . . 4 𝑀 = OrdIso(𝑅, 𝑋)
21oif 8642 . . 3 𝑀:dom 𝑀𝑋
3 ffn 6223 . . 3 (𝑀:dom 𝑀𝑋𝑀 Fn dom 𝑀)
42, 3mp1i 13 . 2 (𝜑𝑀 Fn dom 𝑀)
5 fpwwe2lem9.n . . . . 5 𝑁 = OrdIso(𝑆, 𝑌)
65oif 8642 . . . 4 𝑁:dom 𝑁𝑌
7 ffn 6223 . . . 4 (𝑁:dom 𝑁𝑌𝑁 Fn dom 𝑁)
86, 7mp1i 13 . . 3 (𝜑𝑁 Fn dom 𝑁)
9 fpwwe2lem9.s . . 3 (𝜑 → dom 𝑀 ⊆ dom 𝑁)
10 fnssres 6182 . . 3 ((𝑁 Fn dom 𝑁 ∧ dom 𝑀 ⊆ dom 𝑁) → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
118, 9, 10syl2anc 579 . 2 (𝜑 → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
121oicl 8641 . . . . . 6 Ord dom 𝑀
13 ordelon 5932 . . . . . 6 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ∈ On)
1412, 13mpan 681 . . . . 5 (𝑤 ∈ dom 𝑀𝑤 ∈ On)
15 eleq1w 2827 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝑀𝑦 ∈ dom 𝑀))
16 fveq2 6375 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑀𝑤) = (𝑀𝑦))
17 fveq2 6375 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑁𝑤) = (𝑁𝑦))
1816, 17eqeq12d 2780 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑀𝑦) = (𝑁𝑦)))
1915, 18imbi12d 335 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)) ↔ (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2019imbi2d 331 . . . . . . 7 (𝑤 = 𝑦 → ((𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))) ↔ (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)))))
21 r19.21v 3107 . . . . . . . . 9 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) ↔ (𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2212a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord dom 𝑀)
23 ordelss 5924 . . . . . . . . . . . . . . . . 17 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2422, 23sylan 575 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2524sselda 3761 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ 𝑦𝑤) → 𝑦 ∈ dom 𝑀)
26 pm2.27 42 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑀 → ((𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑦) = (𝑁𝑦)))
2725, 26syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ dom 𝑀) ∧ 𝑦𝑤) → ((𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑦) = (𝑁𝑦)))
2827ralimdva 3109 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
29 fnssres 6182 . . . . . . . . . . . . . . . . 17 ((𝑀 Fn dom 𝑀𝑤 ⊆ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
304, 24, 29syl2an2r 675 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
319adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → dom 𝑀 ⊆ dom 𝑁)
3224, 31sstrd 3771 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑁)
33 fnssres 6182 . . . . . . . . . . . . . . . . 17 ((𝑁 Fn dom 𝑁𝑤 ⊆ dom 𝑁) → (𝑁𝑤) Fn 𝑤)
348, 32, 33syl2an2r 675 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) Fn 𝑤)
35 eqfnfv 6501 . . . . . . . . . . . . . . . 16 (((𝑀𝑤) Fn 𝑤 ∧ (𝑁𝑤) Fn 𝑤) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
3630, 34, 35syl2anc 579 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
37 fvres 6394 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑀𝑤)‘𝑦) = (𝑀𝑦))
38 fvres 6394 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑁𝑤)‘𝑦) = (𝑁𝑦))
3937, 38eqeq12d 2780 . . . . . . . . . . . . . . . 16 (𝑦𝑤 → (((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ (𝑀𝑦) = (𝑁𝑦)))
4039ralbiia 3126 . . . . . . . . . . . . . . 15 (∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦))
4136, 40syl6bb 278 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
42 fpwwe2.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
43 fpwwe2.2 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ V)
4443ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝐴 ∈ V)
45 simpll 783 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝜑)
46 fpwwe2.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4745, 46sylan 575 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
48 fpwwe2lem9.x . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋𝑊𝑅)
4948ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑋𝑊𝑅)
50 fpwwe2lem9.y . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌𝑊𝑆)
5150ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑌𝑊𝑆)
52 simplr 785 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑀)
539sselda 3761 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ∈ dom 𝑁)
5453adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑁)
55 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
5642, 44, 47, 49, 51, 1, 5, 52, 54, 55fpwwe2lem7 9711 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑦𝑆(𝑁𝑤) ∧ (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧))))
5756simpld 488 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → 𝑦𝑆(𝑁𝑤))
5855eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) = (𝑀𝑤))
5942, 44, 47, 51, 49, 5, 1, 54, 52, 58fpwwe2lem7 9711 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ∧ (𝑧𝑆(𝑁𝑤) → (𝑦𝑆𝑧𝑦𝑅𝑧))))
6059simpld 488 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → 𝑦𝑅(𝑀𝑤))
6157, 60impbida 835 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ↔ 𝑦𝑆(𝑁𝑤)))
62 fvex 6388 . . . . . . . . . . . . . . . . . . . 20 (𝑀𝑤) ∈ V
63 vex 3353 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
6463eliniseg 5676 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑤) ∈ V → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤)))
6562, 64ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤))
66 fvex 6388 . . . . . . . . . . . . . . . . . . . 20 (𝑁𝑤) ∈ V
6763eliniseg 5676 . . . . . . . . . . . . . . . . . . . 20 ((𝑁𝑤) ∈ V → (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤)))
6866, 67ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤))
6961, 65, 683bitr4g 305 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦 ∈ (𝑆 “ {(𝑁𝑤)})))
7069eqrdv 2763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 “ {(𝑀𝑤)}) = (𝑆 “ {(𝑁𝑤)}))
71 relinxp 5406 . . . . . . . . . . . . . . . . . . 19 Rel (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
72 relinxp 5406 . . . . . . . . . . . . . . . . . . 19 Rel (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
73 vex 3353 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 ∈ V
7473eliniseg 5676 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀𝑤) ∈ V → (𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑧𝑅(𝑀𝑤)))
7564, 74anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀𝑤) ∈ V → ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))))
7662, 75ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤)))
7756simprd 489 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧)))
7877impr 446 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
7976, 78sylan2b 587 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
8079pm5.32da 574 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧)))
81 df-br 4810 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
82 brinxp2 5348 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
8381, 82bitr3i 268 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
84 df-br 4810 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
85 brinxp2 5348 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8684, 85bitr3i 268 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8780, 83, 863bitr4g 305 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
8871, 72, 87eqrelrdv 5385 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
8970sqxpeqd 5309 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) = ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))
9089ineq2d 3976 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9188, 90eqtrd 2799 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9270, 91oveq12d 6860 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))))
932ffvelrni 6548 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑀 → (𝑀𝑤) ∈ 𝑋)
9493adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) ∈ 𝑋)
9594adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) ∈ 𝑋)
9642, 43, 48fpwwe2lem3 9708 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑀𝑤) ∈ 𝑋) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
9745, 95, 96syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
986ffvelrni 6548 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑁 → (𝑁𝑤) ∈ 𝑌)
9953, 98syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) ∈ 𝑌)
10099adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) ∈ 𝑌)
10142, 43, 50fpwwe2lem3 9708 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑁𝑤) ∈ 𝑌) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10245, 100, 101syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10392, 97, 1023eqtr3d 2807 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
104103ex 401 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) → (𝑀𝑤) = (𝑁𝑤)))
10541, 104sylbird 251 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦) → (𝑀𝑤) = (𝑁𝑤)))
10628, 105syld 47 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤)))
107106ex 401 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ dom 𝑀 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤))))
108107com23 86 . . . . . . . . . 10 (𝜑 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
109108a2i 14 . . . . . . . . 9 ((𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
11021, 109sylbi 208 . . . . . . . 8 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
111110a1i 11 . . . . . . 7 (𝑤 ∈ On → (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))))
11220, 111tfis2 7254 . . . . . 6 (𝑤 ∈ On → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
113112com3l 89 . . . . 5 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑤 ∈ On → (𝑀𝑤) = (𝑁𝑤))))
11414, 113mpdi 45 . . . 4 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))
115114imp 395 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = (𝑁𝑤))
116 fvres 6394 . . . 4 (𝑤 ∈ dom 𝑀 → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
117116adantl 473 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
118115, 117eqtr4d 2802 . 2 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = ((𝑁 ↾ dom 𝑀)‘𝑤))
1194, 11, 118eqfnfvd 6504 1 (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  [wsbc 3596  cin 3731  wss 3732  {csn 4334  cop 4340   class class class wbr 4809  {copab 4871   We wwe 5235   × cxp 5275  ccnv 5276  dom cdm 5277  cres 5279  cima 5280  Ord word 5907  Oncon0 5908   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  OrdIsocoi 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-wrecs 7610  df-recs 7672  df-oi 8622
This theorem is referenced by:  fpwwe2lem9  9713
  Copyright terms: Public domain W3C validator