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Theorem fpwwe2lem8 9444
Description: Lemma for fpwwe2 9450. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
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 8420 . . 3 𝑀:dom 𝑀𝑋
3 ffn 6032 . . 3 (𝑀:dom 𝑀𝑋𝑀 Fn dom 𝑀)
42, 3mp1i 13 . 2 (𝜑𝑀 Fn dom 𝑀)
5 fpwwe2lem9.n . . . . 5 𝑁 = OrdIso(𝑆, 𝑌)
65oif 8420 . . . 4 𝑁:dom 𝑁𝑌
7 ffn 6032 . . . 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 5992 . . 3 ((𝑁 Fn dom 𝑁 ∧ dom 𝑀 ⊆ dom 𝑁) → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
118, 9, 10syl2anc 692 . 2 (𝜑 → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
121oicl 8419 . . . . . 6 Ord dom 𝑀
13 ordelon 5735 . . . . . 6 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ∈ On)
1412, 13mpan 705 . . . . 5 (𝑤 ∈ dom 𝑀𝑤 ∈ On)
15 eleq1 2687 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝑀𝑦 ∈ dom 𝑀))
16 fveq2 6178 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑀𝑤) = (𝑀𝑦))
17 fveq2 6178 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑁𝑤) = (𝑁𝑦))
1816, 17eqeq12d 2635 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑀𝑦) = (𝑁𝑦)))
1915, 18imbi12d 334 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)) ↔ (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2019imbi2d 330 . . . . . . 7 (𝑤 = 𝑦 → ((𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))) ↔ (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)))))
21 r19.21v 2957 . . . . . . . . 9 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) ↔ (𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2212a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord dom 𝑀)
23 ordelss 5727 . . . . . . . . . . . . . . . . 17 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2422, 23sylan 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2524sselda 3595 . . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
294adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑀 Fn dom 𝑀)
30 fnssres 5992 . . . . . . . . . . . . . . . . 17 ((𝑀 Fn dom 𝑀𝑤 ⊆ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
3129, 24, 30syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
328adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑁 Fn dom 𝑁)
339adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → dom 𝑀 ⊆ dom 𝑁)
3424, 33sstrd 3605 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑁)
35 fnssres 5992 . . . . . . . . . . . . . . . . 17 ((𝑁 Fn dom 𝑁𝑤 ⊆ dom 𝑁) → (𝑁𝑤) Fn 𝑤)
3632, 34, 35syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) Fn 𝑤)
37 eqfnfv 6297 . . . . . . . . . . . . . . . 16 (((𝑀𝑤) Fn 𝑤 ∧ (𝑁𝑤) Fn 𝑤) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
3831, 36, 37syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
39 fvres 6194 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑀𝑤)‘𝑦) = (𝑀𝑦))
40 fvres 6194 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑁𝑤)‘𝑦) = (𝑁𝑦))
4139, 40eqeq12d 2635 . . . . . . . . . . . . . . . 16 (𝑦𝑤 → (((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ (𝑀𝑦) = (𝑁𝑦)))
4241ralbiia 2976 . . . . . . . . . . . . . . 15 (∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦))
4338, 42syl6bb 276 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
44 fpwwe2.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
45 fpwwe2.2 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ V)
4645ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝐴 ∈ V)
47 simpll 789 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝜑)
48 fpwwe2.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4947, 48sylan 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
50 fpwwe2lem9.x . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋𝑊𝑅)
5150ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑋𝑊𝑅)
52 fpwwe2lem9.y . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌𝑊𝑆)
5352ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑌𝑊𝑆)
54 simplr 791 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑀)
559sselda 3595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ∈ dom 𝑁)
5655adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑁)
57 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
5844, 46, 49, 51, 53, 1, 5, 54, 56, 57fpwwe2lem7 9443 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑦𝑆(𝑁𝑤) ∧ (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧))))
5958simpld 475 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → 𝑦𝑆(𝑁𝑤))
6057eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) = (𝑀𝑤))
6144, 46, 49, 53, 51, 5, 1, 56, 54, 60fpwwe2lem7 9443 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ∧ (𝑧𝑆(𝑁𝑤) → (𝑦𝑆𝑧𝑦𝑅𝑧))))
6261simpld 475 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → 𝑦𝑅(𝑀𝑤))
6359, 62impbida 876 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ↔ 𝑦𝑆(𝑁𝑤)))
64 fvex 6188 . . . . . . . . . . . . . . . . . . . 20 (𝑀𝑤) ∈ V
65 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
6665eliniseg 5482 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑤) ∈ V → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤)))
6764, 66ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤))
68 fvex 6188 . . . . . . . . . . . . . . . . . . . 20 (𝑁𝑤) ∈ V
6965eliniseg 5482 . . . . . . . . . . . . . . . . . . . 20 ((𝑁𝑤) ∈ V → (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤)))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤))
7163, 67, 703bitr4g 303 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦 ∈ (𝑆 “ {(𝑁𝑤)})))
7271eqrdv 2618 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 “ {(𝑀𝑤)}) = (𝑆 “ {(𝑁𝑤)}))
73 inss2 3826 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ⊆ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))
74 relxp 5217 . . . . . . . . . . . . . . . . . . . 20 Rel ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))
75 relss 5196 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ⊆ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) → (Rel ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) → Rel (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
7673, 74, 75mp2 9 . . . . . . . . . . . . . . . . . . 19 Rel (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
77 inss2 3826 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ⊆ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))
78 relss 5196 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ⊆ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) → (Rel ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) → Rel (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
7977, 74, 78mp2 9 . . . . . . . . . . . . . . . . . . 19 Rel (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
80 vex 3198 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 ∈ V
8180eliniseg 5482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀𝑤) ∈ V → (𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑧𝑅(𝑀𝑤)))
8266, 81anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀𝑤) ∈ V → ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))))
8364, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤)))
8458simprd 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧)))
8584impr 648 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
8683, 85sylan2b 492 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
8786pm5.32da 672 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧)))
88 brinxp2 5170 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑦𝑅𝑧))
89 df-br 4645 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
90 df-3an 1038 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
9188, 89, 903bitr3i 290 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
92 brinxp2 5170 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑦𝑆𝑧))
93 df-br 4645 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
94 df-3an 1038 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑦𝑆𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
9592, 93, 943bitr3i 290 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
9687, 91, 953bitr4g 303 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
9776, 79, 96eqrelrdv 5206 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
9872sqxpeqd 5131 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) = ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))
9998ineq2d 3806 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
10097, 99eqtrd 2654 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
10172, 100oveq12d 6653 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))))
1022ffvelrni 6344 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑀 → (𝑀𝑤) ∈ 𝑋)
103102adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) ∈ 𝑋)
104103adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) ∈ 𝑋)
10544, 45, 50fpwwe2lem3 9440 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑀𝑤) ∈ 𝑋) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
10647, 104, 105syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
1076ffvelrni 6344 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑁 → (𝑁𝑤) ∈ 𝑌)
10855, 107syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) ∈ 𝑌)
109108adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) ∈ 𝑌)
11044, 45, 52fpwwe2lem3 9440 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑁𝑤) ∈ 𝑌) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
11147, 109, 110syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
112101, 106, 1113eqtr3d 2662 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
113112ex 450 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) → (𝑀𝑤) = (𝑁𝑤)))
11443, 113sylbird 250 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦) → (𝑀𝑤) = (𝑁𝑤)))
11528, 114syld 47 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤)))
116115ex 450 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ dom 𝑀 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤))))
117116com23 86 . . . . . . . . . 10 (𝜑 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
118117a2i 14 . . . . . . . . 9 ((𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
11921, 118sylbi 207 . . . . . . . 8 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
120119a1i 11 . . . . . . 7 (𝑤 ∈ On → (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))))
12120, 120tfis2 7041 . . . . . 6 (𝑤 ∈ On → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
122121com3l 89 . . . . 5 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑤 ∈ On → (𝑀𝑤) = (𝑁𝑤))))
12314, 122mpdi 45 . . . 4 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))
124123imp 445 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = (𝑁𝑤))
125 fvres 6194 . . . 4 (𝑤 ∈ dom 𝑀 → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
126125adantl 482 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
127124, 126eqtr4d 2657 . 2 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = ((𝑁 ↾ dom 𝑀)‘𝑤))
1284, 11, 127eqfnfvd 6300 1 (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  [wsbc 3429  cin 3566  wss 3567  {csn 4168  cop 4174   class class class wbr 4644  {copab 4703   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  cres 5106  cima 5107  Rel wrel 5109  Ord word 5710  Oncon0 5711   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  OrdIsocoi 8399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-wrecs 7392  df-recs 7453  df-oi 8400
This theorem is referenced by:  fpwwe2lem9  9445
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