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Theorem isomin 6487
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 3889 . . . 4 (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})))
2 ssel 3562 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
3 isof1o 6473 . . . . . . . . . . . . . . 15 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6051 . . . . . . . . . . . . . . 15 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 fnbrfvb 6146 . . . . . . . . . . . . . . . 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 76 . . . . . . . . . . . . 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 3031 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐶 (𝐻𝑥) = 𝑦 ↔ ∃𝑥𝐶 𝑥𝐻𝑦))
11 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
1211elima 5390 . . . . . . . . . . 11 (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 𝑥𝐻𝑦)
1310, 12syl6rbbr 278 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 (𝐻𝑥) = 𝑦))
14 fvex 6113 . . . . . . . . . . 11 (𝐻𝐷) ∈ V
1511eliniseg 5413 . . . . . . . . . . 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 743 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → ((𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3758 . . . . . . . . 9 (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
19 r19.41v 3070 . . . . . . . . 9 (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
2017, 18, 193bitr4g 302 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
2120adantrr 749 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
22 breq1 4586 . . . . . . . . . . . . . 14 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
2322biimpar 501 . . . . . . . . . . . . 13 (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → (𝐻𝑥)𝑆(𝐻𝐷))
24 vex 3176 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2524eliniseg 5413 . . . . . . . . . . . . . . 15 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2625ad2antll 761 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27 isorel 6476 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2826, 27bitrd 267 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2923, 28syl5ibr 235 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))
3029exp32 629 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))))
312, 30syl9r 76 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))))
3231com34 89 . . . . . . . . 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 2998 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
3521, 34sylbid 229 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
36 elin 3758 . . . . . . . 8 (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
3736exbii 1764 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
38 neq0 3889 . . . . . . 7 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})))
39 df-rex 2902 . . . . . . 7 (∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
4037, 38, 393bitr4i 291 . . . . . 6 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}))
4135, 40syl6ibr 241 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4241exlimdv 1848 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
431, 42syl5bi 231 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4443con4d 113 . 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 6400 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝐶𝐴𝑥𝐶) → (𝐻𝑥) ∈ (𝐻𝐶))
47463expia 1259 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝐶𝐴) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4847adantrr 749 . . . . . . . . 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 218 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥)𝑆(𝐻𝐷)))
52 fvex 6113 . . . . . . . . . . . . . . . 16 (𝐻𝑥) ∈ V
5352eliniseg 5413 . . . . . . . . . . . . . . 15 ((𝐻𝐷) ∈ V → ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
5414, 53ax-mp 5 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷))
5551, 54syl6ibr 241 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5626, 55sylbid 229 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5756exp32 629 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))))
582, 57syl9r 76 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))))
5958com34 89 . . . . . . . . 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 3758 . . . . . 6 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6462, 36, 633imtr4g 284 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)}))))
65 n0i 3879 . . . . 5 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅)
6664, 65syl6 34 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6766exlimdv 1848 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6838, 67syl5bi 231 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6944, 68impcon4bid 216 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 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wrex 2897  Vcvv 3173  cin 3539  wss 3540  c0 3874  {csn 4125   class class class wbr 4583  ccnv 5037  cima 5041   Fn wfn 5799  1-1-ontowf1o 5803  cfv 5804   Isom wiso 5805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812  df-isom 5813
This theorem is referenced by:  isofrlem  6490
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