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Theorem isomin 7084
 Description: Isomorphisms preserve minimal elements. Note that (◡𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥 ∣ 𝑥𝑅𝐷}. 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 4244 . . . 4 (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})))
2 vex 3413 . . . . . . . . . . . 12 𝑦 ∈ V
32elima 5906 . . . . . . . . . . 11 (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 𝑥𝐻𝑦)
4 ssel 3885 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
5 isof1o 7070 . . . . . . . . . . . . . . 15 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
6 f1ofn 6603 . . . . . . . . . . . . . . 15 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
7 fnbrfvb 6706 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
87ex 416 . . . . . . . . . . . . . . 15 (𝐻 Fn 𝐴 → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
95, 6, 83syl 18 . . . . . . . . . . . . . 14 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
104, 9syl9r 78 . . . . . . . . . . . . 13 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))))
1110imp31 421 . . . . . . . . . . . 12 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐶) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
1211rexbidva 3220 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐶 (𝐻𝑥) = 𝑦 ↔ ∃𝑥𝐶 𝑥𝐻𝑦))
133, 12bitr4id 293 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 (𝐻𝑥) = 𝑦))
14 fvex 6671 . . . . . . . . . . 11 (𝐻𝐷) ∈ V
152eliniseg 5930 . . . . . . . . . . 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 633 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → ((𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3874 . . . . . . . . 9 (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
19 r19.41v 3265 . . . . . . . . 9 (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
2017, 18, 193bitr4g 317 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
2120adantrr 716 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
22 breq1 5035 . . . . . . . . . . . . . 14 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
2322biimpar 481 . . . . . . . . . . . . 13 (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → (𝐻𝑥)𝑆(𝐻𝐷))
24 vex 3413 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2524eliniseg 5930 . . . . . . . . . . . . . . 15 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2625ad2antll 728 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27 isorel 7073 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2826, 27bitrd 282 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2923, 28syl5ibr 249 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))
3029exp32 424 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))))
314, 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 422 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))
3433reximdvai 3196 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
3521, 34sylbid 243 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
36 elin 3874 . . . . . . . 8 (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
3736exbii 1849 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
38 neq0 4244 . . . . . . 7 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})))
39 df-rex 3076 . . . . . . 7 (∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
4037, 38, 393bitr4i 306 . . . . . 6 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}))
4135, 40syl6ibr 255 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4241exlimdv 1934 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
431, 42syl5bi 245 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4443con4d 115 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
455, 6syl 17 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
46 fnfvima 6987 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝐶𝐴𝑥𝐶) → (𝐻𝑥) ∈ (𝐻𝐶))
47463expia 1118 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝐶𝐴) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4847adantrr 716 . . . . . . . . 9 ((𝐻 Fn 𝐴 ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4945, 48sylan 583 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
5049adantrd 495 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝐻𝐶)))
5127biimpd 232 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥)𝑆(𝐻𝐷)))
52 fvex 6671 . . . . . . . . . . . . . . . 16 (𝐻𝑥) ∈ V
5352eliniseg 5930 . . . . . . . . . . . . . . 15 ((𝐻𝐷) ∈ V → ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
5414, 53ax-mp 5 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷))
5551, 54syl6ibr 255 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5626, 55sylbid 243 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5756exp32 424 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))))
584, 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 422 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
6160impd 414 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6250, 61jcad 516 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
63 elin 3874 . . . . . 6 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6462, 36, 633imtr4g 299 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)}))))
65 n0i 4232 . . . . 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 1934 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6838, 67syl5bi 245 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6944, 68impcon4bid 230 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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {csn 4522   class class class wbr 5032  ◡ccnv 5523   “ cima 5527   Fn wfn 6330  –1-1-onto→wf1o 6334  ‘cfv 6335   Isom wiso 6336 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-f1o 6342  df-fv 6343  df-isom 6344 This theorem is referenced by:  isofrlem  7087
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