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Theorem f1oiso 5821
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 109 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 5456 . . 3 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
3 df-br 4001 . . . . 5 ((𝐻𝑣)𝑆(𝐻𝑢) ↔ ⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆)
4 eleq2 2241 . . . . . . 7 (𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆 ↔ ⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}))
5 f1fn 5419 . . . . . . . . 9 (𝐻:𝐴1-1𝐵𝐻 Fn 𝐴)
6 funfvex 5528 . . . . . . . . . . . 12 ((Fun 𝐻𝑣 ∈ dom 𝐻) → (𝐻𝑣) ∈ V)
76funfni 5312 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝑣𝐴) → (𝐻𝑣) ∈ V)
8 funfvex 5528 . . . . . . . . . . . 12 ((Fun 𝐻𝑢 ∈ dom 𝐻) → (𝐻𝑢) ∈ V)
98funfni 5312 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝑢𝐴) → (𝐻𝑢) ∈ V)
107, 9anim12dan 600 . . . . . . . . . 10 ((𝐻 Fn 𝐴 ∧ (𝑣𝐴𝑢𝐴)) → ((𝐻𝑣) ∈ V ∧ (𝐻𝑢) ∈ V))
11 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑧 = (𝐻𝑣) → (𝑧 = (𝐻𝑥) ↔ (𝐻𝑣) = (𝐻𝑥)))
1211anbi1d 465 . . . . . . . . . . . . 13 (𝑧 = (𝐻𝑣) → ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦))))
1312anbi1d 465 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑣) → (((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
14132rexbidv 2502 . . . . . . . . . . 11 (𝑧 = (𝐻𝑣) → (∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
15 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑤 = (𝐻𝑢) → (𝑤 = (𝐻𝑦) ↔ (𝐻𝑢) = (𝐻𝑦)))
1615anbi2d 464 . . . . . . . . . . . . 13 (𝑤 = (𝐻𝑢) → (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦))))
1716anbi1d 465 . . . . . . . . . . . 12 (𝑤 = (𝐻𝑢) → ((((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
18172rexbidv 2502 . . . . . . . . . . 11 (𝑤 = (𝐻𝑢) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
1914, 18opelopabg 4265 . . . . . . . . . 10 (((𝐻𝑣) ∈ V ∧ (𝐻𝑢) ∈ V) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
2010, 19syl 14 . . . . . . . . 9 ((𝐻 Fn 𝐴 ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
215, 20sylan 283 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ ∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)))
22 anass 401 . . . . . . . . . . . . . . 15 ((((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ((𝐻𝑣) = (𝐻𝑥) ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)))
23 f1fveq 5767 . . . . . . . . . . . . . . . . . 18 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑥𝐴)) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑣 = 𝑥))
24 equcom 1706 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑥𝑥 = 𝑣)
2523, 24bitrdi 196 . . . . . . . . . . . . . . . . 17 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑥𝐴)) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑥 = 𝑣))
2625anassrs 400 . . . . . . . . . . . . . . . 16 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → ((𝐻𝑣) = (𝐻𝑥) ↔ 𝑥 = 𝑣))
2726anbi1d 465 . . . . . . . . . . . . . . 15 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (((𝐻𝑣) = (𝐻𝑥) ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
2822, 27bitrid 192 . . . . . . . . . . . . . 14 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → ((((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
2928rexbidv 2478 . . . . . . . . . . . . 13 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (∃𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
30 r19.42v 2634 . . . . . . . . . . . . 13 (∃𝑦𝐴 (𝑥 = 𝑣 ∧ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)))
3129, 30bitrdi 196 . . . . . . . . . . . 12 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ 𝑥𝐴) → (∃𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
3231rexbidva 2474 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦))))
33 breq1 4003 . . . . . . . . . . . . . . 15 (𝑥 = 𝑣 → (𝑥𝑅𝑦𝑣𝑅𝑦))
3433anbi2d 464 . . . . . . . . . . . . . 14 (𝑥 = 𝑣 → (((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦) ↔ ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3534rexbidv 2478 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3635ceqsrexv 2867 . . . . . . . . . . . 12 (𝑣𝐴 → (∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3736adantl 277 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴 (𝑥 = 𝑣 ∧ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑥𝑅𝑦)) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
3832, 37bitrd 188 . . . . . . . . . 10 ((𝐻:𝐴1-1𝐵𝑣𝐴) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ ∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦)))
39 f1fveq 5767 . . . . . . . . . . . . . . 15 ((𝐻:𝐴1-1𝐵 ∧ (𝑢𝐴𝑦𝐴)) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑢 = 𝑦))
40 equcom 1706 . . . . . . . . . . . . . . 15 (𝑢 = 𝑦𝑦 = 𝑢)
4139, 40bitrdi 196 . . . . . . . . . . . . . 14 ((𝐻:𝐴1-1𝐵 ∧ (𝑢𝐴𝑦𝐴)) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑦 = 𝑢))
4241anassrs 400 . . . . . . . . . . . . 13 (((𝐻:𝐴1-1𝐵𝑢𝐴) ∧ 𝑦𝐴) → ((𝐻𝑢) = (𝐻𝑦) ↔ 𝑦 = 𝑢))
4342anbi1d 465 . . . . . . . . . . . 12 (((𝐻:𝐴1-1𝐵𝑢𝐴) ∧ 𝑦𝐴) → (((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ (𝑦 = 𝑢𝑣𝑅𝑦)))
4443rexbidva 2474 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ ∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦)))
45 breq2 4004 . . . . . . . . . . . . 13 (𝑦 = 𝑢 → (𝑣𝑅𝑦𝑣𝑅𝑢))
4645ceqsrexv 2867 . . . . . . . . . . . 12 (𝑢𝐴 → (∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4746adantl 277 . . . . . . . . . . 11 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 (𝑦 = 𝑢𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4844, 47bitrd 188 . . . . . . . . . 10 ((𝐻:𝐴1-1𝐵𝑢𝐴) → (∃𝑦𝐴 ((𝐻𝑢) = (𝐻𝑦) ∧ 𝑣𝑅𝑦) ↔ 𝑣𝑅𝑢))
4938, 48sylan9bb 462 . . . . . . . . 9 (((𝐻:𝐴1-1𝐵𝑣𝐴) ∧ (𝐻:𝐴1-1𝐵𝑢𝐴)) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ 𝑣𝑅𝑢))
5049anandis 592 . . . . . . . 8 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) → (∃𝑥𝐴𝑦𝐴 (((𝐻𝑣) = (𝐻𝑥) ∧ (𝐻𝑢) = (𝐻𝑦)) ∧ 𝑥𝑅𝑦) ↔ 𝑣𝑅𝑢))
5121, 50bitrd 188 . . . . . . 7 ((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)} ↔ 𝑣𝑅𝑢))
524, 51sylan9bbr 463 . . . . . 6 (((𝐻:𝐴1-1𝐵 ∧ (𝑣𝐴𝑢𝐴)) ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆𝑣𝑅𝑢))
5352an32s 568 . . . . 5 (((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) ∧ (𝑣𝐴𝑢𝐴)) → (⟨(𝐻𝑣), (𝐻𝑢)⟩ ∈ 𝑆𝑣𝑅𝑢))
543, 53bitr2id 193 . . . 4 (((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) ∧ (𝑣𝐴𝑢𝐴)) → (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
5554ralrimivva 2559 . . 3 ((𝐻:𝐴1-1𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
562, 55sylan 283 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢)))
57 df-isom 5221 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑣𝐴𝑢𝐴 (𝑣𝑅𝑢 ↔ (𝐻𝑣)𝑆(𝐻𝑢))))
581, 56, 57sylanbrc 417 1 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  cop 3594   class class class wbr 4000  {copab 4060   Fn wfn 5207  1-1wf1 5209  1-1-ontowf1o 5211  cfv 5212   Isom wiso 5213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-isom 5221
This theorem is referenced by:  f1oiso2  5822
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