MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe2lem7 Structured version   Visualization version   GIF version

Theorem fpwwe2lem7 10632
Description: Lemma for fpwwe2 10638. 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⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴𝑉)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem8.x (𝜑𝑋𝑊𝑅)
fpwwe2lem8.y (𝜑𝑌𝑊𝑆)
fpwwe2lem8.m 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem8.n 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem8.s (𝜑 → dom 𝑀 ⊆ dom 𝑁)
Assertion
Ref Expression
fpwwe2lem7 (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)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem7
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2lem8.m . . . 4 𝑀 = OrdIso(𝑅, 𝑋)
21oif 9525 . . 3 𝑀:dom 𝑀𝑋
3 ffn 6718 . . 3 (𝑀:dom 𝑀𝑋𝑀 Fn dom 𝑀)
42, 3mp1i 13 . 2 (𝜑𝑀 Fn dom 𝑀)
5 fpwwe2lem8.n . . . . 5 𝑁 = OrdIso(𝑆, 𝑌)
65oif 9525 . . . 4 𝑁:dom 𝑁𝑌
7 ffn 6718 . . . 4 (𝑁:dom 𝑁𝑌𝑁 Fn dom 𝑁)
86, 7mp1i 13 . . 3 (𝜑𝑁 Fn dom 𝑁)
9 fpwwe2lem8.s . . 3 (𝜑 → dom 𝑀 ⊆ dom 𝑁)
108, 9fnssresd 6675 . 2 (𝜑 → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
111oicl 9524 . . . . . 6 Ord dom 𝑀
12 ordelon 6389 . . . . . 6 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ∈ On)
1311, 12mpan 689 . . . . 5 (𝑤 ∈ dom 𝑀𝑤 ∈ On)
14 eleq1w 2817 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝑀𝑦 ∈ dom 𝑀))
15 fveq2 6892 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑀𝑤) = (𝑀𝑦))
16 fveq2 6892 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑁𝑤) = (𝑁𝑦))
1715, 16eqeq12d 2749 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑀𝑦) = (𝑁𝑦)))
1814, 17imbi12d 345 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)) ↔ (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
1918imbi2d 341 . . . . . . 7 (𝑤 = 𝑦 → ((𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))) ↔ (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)))))
20 r19.21v 3180 . . . . . . . . 9 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) ↔ (𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2111a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord dom 𝑀)
22 ordelss 6381 . . . . . . . . . . . . . . . . 17 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2321, 22sylan 581 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2423sselda 3983 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ 𝑦𝑤) → 𝑦 ∈ dom 𝑀)
25 pm2.27 42 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑀 → ((𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑦) = (𝑁𝑦)))
2624, 25syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ dom 𝑀) ∧ 𝑦𝑤) → ((𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑦) = (𝑁𝑦)))
2726ralimdva 3168 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
28 fnssres 6674 . . . . . . . . . . . . . . . . 17 ((𝑀 Fn dom 𝑀𝑤 ⊆ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
294, 23, 28syl2an2r 684 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
309adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → dom 𝑀 ⊆ dom 𝑁)
3123, 30sstrd 3993 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑁)
32 fnssres 6674 . . . . . . . . . . . . . . . . 17 ((𝑁 Fn dom 𝑁𝑤 ⊆ dom 𝑁) → (𝑁𝑤) Fn 𝑤)
338, 31, 32syl2an2r 684 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) Fn 𝑤)
34 eqfnfv 7033 . . . . . . . . . . . . . . . 16 (((𝑀𝑤) Fn 𝑤 ∧ (𝑁𝑤) Fn 𝑤) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
3529, 33, 34syl2anc 585 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
36 fvres 6911 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑀𝑤)‘𝑦) = (𝑀𝑦))
37 fvres 6911 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑁𝑤)‘𝑦) = (𝑁𝑦))
3836, 37eqeq12d 2749 . . . . . . . . . . . . . . . 16 (𝑦𝑤 → (((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ (𝑀𝑦) = (𝑁𝑦)))
3938ralbiia 3092 . . . . . . . . . . . . . . 15 (∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦))
4035, 39bitrdi 287 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
41 fpwwe2.1 . . . . . . . . . . . . . . . . . . . . . 22 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
42 fpwwe2.2 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴𝑉)
4342ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝐴𝑉)
44 simpll 766 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝜑)
45 fpwwe2.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4644, 45sylan 581 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
47 fpwwe2lem8.x . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋𝑊𝑅)
4847ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑋𝑊𝑅)
49 fpwwe2lem8.y . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌𝑊𝑆)
5049ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑌𝑊𝑆)
51 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑀)
529sselda 3983 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ∈ dom 𝑁)
5352adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑁)
54 simpr 486 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
5541, 43, 46, 48, 50, 1, 5, 51, 53, 54fpwwe2lem6 10631 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑦𝑆(𝑁𝑤) ∧ (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧))))
5655simpld 496 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → 𝑦𝑆(𝑁𝑤))
5754eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) = (𝑀𝑤))
5841, 43, 46, 50, 48, 5, 1, 53, 51, 57fpwwe2lem6 10631 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ∧ (𝑧𝑆(𝑁𝑤) → (𝑦𝑆𝑧𝑦𝑅𝑧))))
5958simpld 496 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → 𝑦𝑅(𝑀𝑤))
6056, 59impbida 800 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ↔ 𝑦𝑆(𝑁𝑤)))
61 fvex 6905 . . . . . . . . . . . . . . . . . . . 20 (𝑀𝑤) ∈ V
62 vex 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
6362eliniseg 6094 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑤) ∈ V → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤)))
6461, 63ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤))
65 fvex 6905 . . . . . . . . . . . . . . . . . . . 20 (𝑁𝑤) ∈ V
6662eliniseg 6094 . . . . . . . . . . . . . . . . . . . 20 ((𝑁𝑤) ∈ V → (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤)))
6765, 66ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤))
6860, 64, 673bitr4g 314 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦 ∈ (𝑆 “ {(𝑁𝑤)})))
6968eqrdv 2731 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 “ {(𝑀𝑤)}) = (𝑆 “ {(𝑁𝑤)}))
70 relinxp 5815 . . . . . . . . . . . . . . . . . . 19 Rel (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
71 relinxp 5815 . . . . . . . . . . . . . . . . . . 19 Rel (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
72 vex 3479 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 ∈ V
7372eliniseg 6094 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀𝑤) ∈ V → (𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑧𝑅(𝑀𝑤)))
7463, 73anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀𝑤) ∈ V → ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))))
7561, 74ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ↔ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤)))
7655simprd 497 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧)))
7776impr 456 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
7875, 77sylan2b 595 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
7978pm5.32da 580 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧)))
80 df-br 5150 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
81 brinxp2 5754 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
8280, 81bitr3i 277 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
83 df-br 5150 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
84 brinxp2 5754 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8583, 84bitr3i 277 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8679, 82, 853bitr4g 314 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
8770, 71, 86eqrelrdv 5793 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
8869sqxpeqd 5709 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) = ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))
8988ineq2d 4213 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9087, 89eqtrd 2773 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9169, 90oveq12d 7427 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))))
922ffvelcdmi 7086 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑀 → (𝑀𝑤) ∈ 𝑋)
9392adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) ∈ 𝑋)
9493adantr 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) ∈ 𝑋)
9541, 42, 47fpwwe2lem3 10628 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑀𝑤) ∈ 𝑋) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
9644, 94, 95syl2anc 585 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
976ffvelcdmi 7086 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑁 → (𝑁𝑤) ∈ 𝑌)
9852, 97syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) ∈ 𝑌)
9998adantr 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) ∈ 𝑌)
10041, 42, 49fpwwe2lem3 10628 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑁𝑤) ∈ 𝑌) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10144, 99, 100syl2anc 585 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10291, 96, 1013eqtr3d 2781 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
103102ex 414 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) → (𝑀𝑤) = (𝑁𝑤)))
10440, 103sylbird 260 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦) → (𝑀𝑤) = (𝑁𝑤)))
10527, 104syld 47 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤)))
106105ex 414 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ dom 𝑀 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑀𝑤) = (𝑁𝑤))))
107106com23 86 . . . . . . . . . 10 (𝜑 → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
108107a2i 14 . . . . . . . . 9 ((𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
10920, 108sylbi 216 . . . . . . . 8 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
110109a1i 11 . . . . . . 7 (𝑤 ∈ On → (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))))
11119, 110tfis2 7846 . . . . . 6 (𝑤 ∈ On → (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))))
112111com3l 89 . . . . 5 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑤 ∈ On → (𝑀𝑤) = (𝑁𝑤))))
11313, 112mpdi 45 . . . 4 (𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)))
114113imp 408 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = (𝑁𝑤))
115 fvres 6911 . . . 4 (𝑤 ∈ dom 𝑀 → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
116115adantl 483 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
117114, 116eqtr4d 2776 . 2 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = ((𝑁 ↾ dom 𝑀)‘𝑤))
1184, 10, 117eqfnfvd 7036 1 (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  [wsbc 3778  cin 3948  wss 3949  {csn 4629  cop 4635   class class class wbr 5149  {copab 5211   We wwe 5631   × cxp 5675  ccnv 5676  dom cdm 5677  cres 5679  cima 5680  Ord word 6364  Oncon0 6365   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  OrdIsocoi 9504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-oi 9505
This theorem is referenced by:  fpwwe2lem8  10633
  Copyright terms: Public domain W3C validator