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Theorem f1oiso 7099
Description: Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
f1oiso ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤
Allowed substitution hints:   𝐵(𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem f1oiso
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 6610 . . 3 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
3 df-br 5063 . . . . 5 ((𝐻𝑣)𝑆(𝐻𝑢) ↔ ⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆)
4 eleq2 2905 . . . . . . 7 (𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆 ↔ ⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}))
5 fvex 6679 . . . . . . . . 9 (𝐻𝑣) ∈ V
6 fvex 6679 . . . . . . . . 9 (𝐻𝑢) ∈ V
7 eqeq1 2829 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑣) → (𝑧 = (𝐻𝑥) ↔ (𝐻𝑣) = (𝐻𝑥)))
87anbi1d 629 . . . . . . . . . . 11 (𝑧 = (𝐻𝑣) → ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦))))
98anbi1d 629 . . . . . . . . . 10 (𝑧 = (𝐻𝑣) → (((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
1092rexbidv 3304 . . . . . . . . 9 (𝑧 = (𝐻𝑣) → (∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
11 eqeq1 2829 . . . . . . . . . . . 12 (𝑤 = (𝐻𝑢) → (𝑤 = (𝐻𝑦) ↔ (𝐻𝑢) = (𝐻𝑦)))
1211anbi2d 628 . . . . . . . . . . 11 (𝑤 = (𝐻𝑢) → (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦))))
1312anbi1d 629 . . . . . . . . . 10 (𝑤 = (𝐻𝑢) → ((((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
14132rexbidv 3304 . . . . . . . . 9 (𝑤 = (𝐻𝑢) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
155, 6, 10, 14opelopab 5425 . . . . . . . 8 (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦))
16 anass 469 . . . . . . . . . . . . . . 15 ((((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)))
17 f1fveq 7017 . . . . . . . . . . . . . . . . . 18 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑥𝐴)) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑣 = 𝑥))
18 equcom 2018 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑥𝑥 = 𝑣)
1917, 18syl6bb 288 . . . . . . . . . . . . . . . . 17 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑥𝐴)) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑥 = 𝑣))
2019anassrs 468 . . . . . . . . . . . . . . . 16 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑥 = 𝑣))
2120anbi1d 629 . . . . . . . . . . . . . . 15 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (((𝐻𝑣) = (𝐻𝑥) ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
2216, 21syl5bb 284 . . . . . . . . . . . . . 14 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → ((((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
2322rexbidv 3301 . . . . . . . . . . . . 13 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (∃𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
24 r19.42v 3354 . . . . . . . . . . . . 13 (∃𝑦𝐴 (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)))
2523, 24syl6bb 288 . . . . . . . . . . . 12 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (∃𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
2625rexbidva 3300 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
27 breq1 5065 . . . . . . . . . . . . . . 15 (𝑥 = 𝑣 → (𝑥𝑅𝑦𝑣𝑅𝑦))
2827anbi2d 628 . . . . . . . . . . . . . 14 (𝑥 = 𝑣 → (((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦) ↔ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
2928rexbidv 3301 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3029ceqsrexv 3652 . . . . . . . . . . . 12 (𝑣𝐴 → (∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3130adantl 482 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3226, 31bitrd 280 . . . . . . . . . 10 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
33 f1fveq 7017 . . . . . . . . . . . . . . 15 ((𝐻:𝐴1-1𝐵 ∧ (𝑢𝐴𝑦𝐴)) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑢 = 𝑦))
34 equcom 2018 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦𝑦 = 𝑢)
3533, 34syl6bb 288 . . . . . . . . . . . . . 14 ((𝐻:𝐴1-1𝐵 ∧ (𝑢𝐴𝑦𝐴)) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑦 = 𝑢))
3635anassrs 468 . . . . . . . . . . . . 13 (((𝐻:𝐴1-1𝐵𝑢𝐴) ∧ 𝑦𝐴) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑦 = 𝑢))
3736anbi1d 629 . . . . . . . . . . . 12 (((𝐻:𝐴1-1𝐵𝑢𝐴) ∧ 𝑦𝐴) → (((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ (𝑦 = 𝑢𝑣𝑅𝑦)))
3837rexbidva 3300 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ ∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦)))
39 breq2 5066 . . . . . . . . . . . . 13 (𝑦 = 𝑢 → (𝑣𝑅𝑦𝑣𝑅𝑢))
4039ceqsrexv 3652 . . . . . . . . . . . 12 (𝑢𝐴 → (∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4140adantl 482 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4238, 41bitrd 280 . . . . . . . . . 10 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4332, 42sylan9bb 510 . . . . . . . . 9 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ (𝐻:𝐴1-1𝐵𝑢𝐴)) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ 𝑣𝑅𝑢))
4443anandis 674 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ 𝑣𝑅𝑢))
4515, 44syl5bb 284 . . . . . . 7 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ 𝑣𝑅𝑢))
464, 45sylan9bbr 511 . . . . . 6 (((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆𝑣𝑅𝑢))
4746an32s 648 . . . . 5 (((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆𝑣𝑅𝑢))
483, 47syl5rbb 285 . . . 4 (((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) ∧ (𝑣𝐴𝑢𝐴)) → (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
4948ralrimivva 3195 . . 3 ((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
502, 49sylan 580 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
51 df-isom 6360 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢))))
521, 50, 51sylanbrc 583 1 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143  cop 4569   class class class wbr 5062  {copab 5124  1-1wf1 6348  1-1-ontowf1o 6350  cfv 6351   Isom wiso 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-f1o 6358  df-fv 6359  df-isom 6360
This theorem is referenced by:  f1oiso2  7100  hartogslem1  8998  cnso  15592
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