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Theorem isomin 6627
Description: Isomorphisms preserve minimal elements. Note that (𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
Assertion
Ref Expression
isomin ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))

Proof of Theorem isomin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 3963 . . . 4 (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})))
2 ssel 3630 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
3 isof1o 6613 . . . . . . . . . . . . . . 15 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6176 . . . . . . . . . . . . . . 15 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 fnbrfvb 6274 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
65ex 449 . . . . . . . . . . . . . . 15 (𝐻 Fn 𝐴 → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
73, 4, 63syl 18 . . . . . . . . . . . . . 14 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
82, 7syl9r 78 . . . . . . . . . . . . 13 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))))
98imp31 447 . . . . . . . . . . . 12 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐶) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
109rexbidva 3078 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐶 (𝐻𝑥) = 𝑦 ↔ ∃𝑥𝐶 𝑥𝐻𝑦))
11 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
1211elima 5506 . . . . . . . . . . 11 (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 𝑥𝐻𝑦)
1310, 12syl6rbbr 279 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 (𝐻𝑥) = 𝑦))
14 fvex 6239 . . . . . . . . . . 11 (𝐻𝐷) ∈ V
1511eliniseg 5529 . . . . . . . . . . 11 ((𝐻𝐷) ∈ V → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1614, 15mp1i 13 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1713, 16anbi12d 747 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → ((𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3829 . . . . . . . . 9 (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
19 r19.41v 3118 . . . . . . . . 9 (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
2017, 18, 193bitr4g 303 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
2120adantrr 753 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
22 breq1 4688 . . . . . . . . . . . . . 14 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
2322biimpar 501 . . . . . . . . . . . . 13 (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → (𝐻𝑥)𝑆(𝐻𝐷))
24 vex 3234 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2524eliniseg 5529 . . . . . . . . . . . . . . 15 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2625ad2antll 765 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27 isorel 6616 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2826, 27bitrd 268 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2923, 28syl5ibr 236 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))
3029exp32 630 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))))
312, 30syl9r 78 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))))
3231com34 91 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝐷𝐴 → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))))
3332imp32 448 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))
3433reximdvai 3044 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
3521, 34sylbid 230 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
36 elin 3829 . . . . . . . 8 (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
3736exbii 1814 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
38 neq0 3963 . . . . . . 7 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})))
39 df-rex 2947 . . . . . . 7 (∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
4037, 38, 393bitr4i 292 . . . . . 6 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}))
4135, 40syl6ibr 242 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4241exlimdv 1901 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
431, 42syl5bi 232 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4443con4d 114 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
453, 4syl 17 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
46 fnfvima 6536 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝐶𝐴𝑥𝐶) → (𝐻𝑥) ∈ (𝐻𝐶))
47463expia 1286 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝐶𝐴) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4847adantrr 753 . . . . . . . . 9 ((𝐻 Fn 𝐴 ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4945, 48sylan 487 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
5049adantrd 483 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝐻𝐶)))
5127biimpd 219 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥)𝑆(𝐻𝐷)))
52 fvex 6239 . . . . . . . . . . . . . . . 16 (𝐻𝑥) ∈ V
5352eliniseg 5529 . . . . . . . . . . . . . . 15 ((𝐻𝐷) ∈ V → ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
5414, 53ax-mp 5 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷))
5551, 54syl6ibr 242 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5626, 55sylbid 230 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5756exp32 630 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))))
582, 57syl9r 78 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))))
5958com34 91 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝐷𝐴 → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))))
6059imp32 448 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
6160impd 446 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6250, 61jcad 554 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
63 elin 3829 . . . . . 6 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6462, 36, 633imtr4g 285 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)}))))
65 n0i 3953 . . . . 5 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅)
6664, 65syl6 35 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6766exlimdv 1901 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6838, 67syl5bi 232 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6944, 68impcon4bid 217 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  ccnv 5142  cima 5146   Fn wfn 5921  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-f1o 5933  df-fv 5934  df-isom 5935
This theorem is referenced by:  isofrlem  6630
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