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Theorem fpwwe2lem7 10634
Description: Lemma for fpwwe2 10640. 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 9527 . . 3 𝑀:dom 𝑀𝑋
3 ffn 6717 . . 3 (𝑀:dom 𝑀𝑋𝑀 Fn dom 𝑀)
42, 3mp1i 13 . 2 (𝜑𝑀 Fn dom 𝑀)
5 fpwwe2lem8.n . . . . 5 𝑁 = OrdIso(𝑆, 𝑌)
65oif 9527 . . . 4 𝑁:dom 𝑁𝑌
7 ffn 6717 . . . 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 6674 . 2 (𝜑 → (𝑁 ↾ dom 𝑀) Fn dom 𝑀)
111oicl 9526 . . . . . 6 Ord dom 𝑀
12 ordelon 6388 . . . . . 6 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ∈ On)
1311, 12mpan 688 . . . . 5 (𝑤 ∈ dom 𝑀𝑤 ∈ On)
14 eleq1w 2816 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝑀𝑦 ∈ dom 𝑀))
15 fveq2 6891 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑀𝑤) = (𝑀𝑦))
16 fveq2 6891 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑁𝑤) = (𝑁𝑦))
1715, 16eqeq12d 2748 . . . . . . . . 9 (𝑤 = 𝑦 → ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑀𝑦) = (𝑁𝑦)))
1814, 17imbi12d 344 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤)) ↔ (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
1918imbi2d 340 . . . . . . 7 (𝑤 = 𝑦 → ((𝜑 → (𝑤 ∈ dom 𝑀 → (𝑀𝑤) = (𝑁𝑤))) ↔ (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)))))
20 r19.21v 3179 . . . . . . . . 9 (∀𝑦𝑤 (𝜑 → (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))) ↔ (𝜑 → ∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦))))
2111a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → Ord dom 𝑀)
22 ordelss 6380 . . . . . . . . . . . . . . . . 17 ((Ord dom 𝑀𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2321, 22sylan 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑀)
2423sselda 3982 . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ dom 𝑀) → (∀𝑦𝑤 (𝑦 ∈ dom 𝑀 → (𝑀𝑦) = (𝑁𝑦)) → ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦)))
28 fnssres 6673 . . . . . . . . . . . . . . . . 17 ((𝑀 Fn dom 𝑀𝑤 ⊆ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
294, 23, 28syl2an2r 683 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) Fn 𝑤)
309adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → dom 𝑀 ⊆ dom 𝑁)
3123, 30sstrd 3992 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ⊆ dom 𝑁)
32 fnssres 6673 . . . . . . . . . . . . . . . . 17 ((𝑁 Fn dom 𝑁𝑤 ⊆ dom 𝑁) → (𝑁𝑤) Fn 𝑤)
338, 31, 32syl2an2r 683 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) Fn 𝑤)
34 eqfnfv 7032 . . . . . . . . . . . . . . . 16 (((𝑀𝑤) Fn 𝑤 ∧ (𝑁𝑤) Fn 𝑤) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
3529, 33, 34syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) ↔ ∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦)))
36 fvres 6910 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑀𝑤)‘𝑦) = (𝑀𝑦))
37 fvres 6910 . . . . . . . . . . . . . . . . 17 (𝑦𝑤 → ((𝑁𝑤)‘𝑦) = (𝑁𝑦))
3836, 37eqeq12d 2748 . . . . . . . . . . . . . . . 16 (𝑦𝑤 → (((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ (𝑀𝑦) = (𝑁𝑦)))
3938ralbiia 3091 . . . . . . . . . . . . . . 15 (∀𝑦𝑤 ((𝑀𝑤)‘𝑦) = ((𝑁𝑤)‘𝑦) ↔ ∀𝑦𝑤 (𝑀𝑦) = (𝑁𝑦))
4035, 39bitrdi 286 . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝐴𝑉)
44 simpll 765 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝜑)
45 fpwwe2.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4644, 45sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
47 fpwwe2lem8.x . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋𝑊𝑅)
4847ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑋𝑊𝑅)
49 fpwwe2lem8.y . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌𝑊𝑆)
5049ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑌𝑊𝑆)
51 simplr 767 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑀)
529sselda 3982 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ dom 𝑀) → 𝑤 ∈ dom 𝑁)
5352adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → 𝑤 ∈ dom 𝑁)
54 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
5541, 43, 46, 48, 50, 1, 5, 51, 53, 54fpwwe2lem6 10633 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑦𝑆(𝑁𝑤) ∧ (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧))))
5655simpld 495 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → 𝑦𝑆(𝑁𝑤))
5754eqcomd 2738 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) = (𝑀𝑤))
5841, 43, 46, 50, 48, 5, 1, 53, 51, 57fpwwe2lem6 10633 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ∧ (𝑧𝑆(𝑁𝑤) → (𝑦𝑆𝑧𝑦𝑅𝑧))))
5958simpld 495 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑆(𝑁𝑤)) → 𝑦𝑅(𝑀𝑤))
6056, 59impbida 799 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦𝑅(𝑀𝑤) ↔ 𝑦𝑆(𝑁𝑤)))
61 fvex 6904 . . . . . . . . . . . . . . . . . . . 20 (𝑀𝑤) ∈ V
62 vex 3478 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
6362eliniseg 6093 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑤) ∈ V → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤)))
6461, 63ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦𝑅(𝑀𝑤))
65 fvex 6904 . . . . . . . . . . . . . . . . . . . 20 (𝑁𝑤) ∈ V
6662eliniseg 6093 . . . . . . . . . . . . . . . . . . . 20 ((𝑁𝑤) ∈ V → (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤)))
6765, 66ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑆 “ {(𝑁𝑤)}) ↔ 𝑦𝑆(𝑁𝑤))
6860, 64, 673bitr4g 313 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑦 ∈ (𝑆 “ {(𝑁𝑤)})))
6968eqrdv 2730 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 “ {(𝑀𝑤)}) = (𝑆 “ {(𝑁𝑤)}))
70 relinxp 5814 . . . . . . . . . . . . . . . . . . 19 Rel (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
71 relinxp 5814 . . . . . . . . . . . . . . . . . . 19 Rel (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))
72 vex 3478 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑧 ∈ V
7372eliniseg 6093 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀𝑤) ∈ V → (𝑧 ∈ (𝑅 “ {(𝑀𝑤)}) ↔ 𝑧𝑅(𝑀𝑤)))
7463, 73anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ 𝑦𝑅(𝑀𝑤)) → (𝑧𝑅(𝑀𝑤) → (𝑦𝑅𝑧𝑦𝑆𝑧)))
7776impr 455 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦𝑅(𝑀𝑤) ∧ 𝑧𝑅(𝑀𝑤))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
7875, 77sylan2b 594 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) ∧ (𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)}))) → (𝑦𝑅𝑧𝑦𝑆𝑧))
7978pm5.32da 579 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧)))
80 df-br 5149 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
81 brinxp2 5753 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
8280, 81bitr3i 276 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑅𝑧))
83 df-br 5149 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
84 brinxp2 5753 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))𝑧 ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8583, 84bitr3i 276 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ((𝑦 ∈ (𝑅 “ {(𝑀𝑤)}) ∧ 𝑧 ∈ (𝑅 “ {(𝑀𝑤)})) ∧ 𝑦𝑆𝑧))
8679, 82, 853bitr4g 313 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (⟨𝑦, 𝑧⟩ ∈ (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) ↔ ⟨𝑦, 𝑧⟩ ∈ (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))))
8770, 71, 86eqrelrdv 5792 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))))
8869sqxpeqd 5708 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})) = ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))
8988ineq2d 4212 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑆 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9087, 89eqtrd 2772 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)}))) = (𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)}))))
9169, 90oveq12d 7429 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))))
922ffvelcdmi 7085 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑀 → (𝑀𝑤) ∈ 𝑋)
9392adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) ∈ 𝑋)
9493adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) ∈ 𝑋)
9541, 42, 47fpwwe2lem3 10630 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑀𝑤) ∈ 𝑋) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
9644, 94, 95syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑅 “ {(𝑀𝑤)})𝐹(𝑅 ∩ ((𝑅 “ {(𝑀𝑤)}) × (𝑅 “ {(𝑀𝑤)})))) = (𝑀𝑤))
976ffvelcdmi 7085 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ dom 𝑁 → (𝑁𝑤) ∈ 𝑌)
9852, 97syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ dom 𝑀) → (𝑁𝑤) ∈ 𝑌)
9998adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑁𝑤) ∈ 𝑌)
10041, 42, 49fpwwe2lem3 10630 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑁𝑤) ∈ 𝑌) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10144, 99, 100syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → ((𝑆 “ {(𝑁𝑤)})𝐹(𝑆 ∩ ((𝑆 “ {(𝑁𝑤)}) × (𝑆 “ {(𝑁𝑤)})))) = (𝑁𝑤))
10291, 96, 1013eqtr3d 2780 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ dom 𝑀) ∧ (𝑀𝑤) = (𝑁𝑤)) → (𝑀𝑤) = (𝑁𝑤))
103102ex 413 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑀𝑤) = (𝑁𝑤) → (𝑀𝑤) = (𝑁𝑤)))
10440, 103sylbird 259 . . . . . . . . . . . . 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 413 . . . . . . . . . . 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 7848 . . . . . 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 407 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = (𝑁𝑤))
115 fvres 6910 . . . 4 (𝑤 ∈ dom 𝑀 → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
116115adantl 482 . . 3 ((𝜑𝑤 ∈ dom 𝑀) → ((𝑁 ↾ dom 𝑀)‘𝑤) = (𝑁𝑤))
117114, 116eqtr4d 2775 . 2 ((𝜑𝑤 ∈ dom 𝑀) → (𝑀𝑤) = ((𝑁 ↾ dom 𝑀)‘𝑤))
1184, 10, 117eqfnfvd 7035 1 (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  [wsbc 3777  cin 3947  wss 3948  {csn 4628  cop 4634   class class class wbr 5148  {copab 5210   We wwe 5630   × cxp 5674  ccnv 5675  dom cdm 5676  cres 5678  cima 5679  Ord word 6363  Oncon0 6364   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  OrdIsocoi 9506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-oi 9507
This theorem is referenced by:  fpwwe2lem8  10635
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