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Theorem isomin 6461
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 3884 . . . 4 (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})))
2 ssel 3557 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
3 isof1o 6447 . . . . . . . . . . . . . . 15 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6032 . . . . . . . . . . . . . . 15 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 fnbrfvb 6127 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
65ex 448 . . . . . . . . . . . . . . 15 (𝐻 Fn 𝐴 → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
73, 4, 63syl 18 . . . . . . . . . . . . . 14 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦)))
82, 7syl9r 75 . . . . . . . . . . . . 13 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))))
98imp31 446 . . . . . . . . . . . 12 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐶) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
109rexbidva 3026 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐶 (𝐻𝑥) = 𝑦 ↔ ∃𝑥𝐶 𝑥𝐻𝑦))
11 vex 3171 . . . . . . . . . . . 12 𝑦 ∈ V
1211elima 5373 . . . . . . . . . . 11 (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 𝑥𝐻𝑦)
1310, 12syl6rbbr 277 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 (𝐻𝑥) = 𝑦))
14 fvex 6094 . . . . . . . . . . 11 (𝐻𝐷) ∈ V
1511eliniseg 5396 . . . . . . . . . . 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 742 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → ((𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3753 . . . . . . . . 9 (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
19 r19.41v 3065 . . . . . . . . 9 (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
2017, 18, 193bitr4g 301 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
2120adantrr 748 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
22 breq1 4576 . . . . . . . . . . . . . 14 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
2322biimpar 500 . . . . . . . . . . . . 13 (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → (𝐻𝑥)𝑆(𝐻𝐷))
24 vex 3171 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2524eliniseg 5396 . . . . . . . . . . . . . . 15 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2625ad2antll 760 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27 isorel 6450 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2826, 27bitrd 266 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2923, 28syl5ibr 234 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))
3029exp32 628 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))))
312, 30syl9r 75 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))))
3231com34 88 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝐷𝐴 → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))))
3332imp32 447 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))
3433reximdvai 2993 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
3521, 34sylbid 228 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
36 elin 3753 . . . . . . . 8 (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
3736exbii 1762 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
38 neq0 3884 . . . . . . 7 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})))
39 df-rex 2897 . . . . . . 7 (∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
4037, 38, 393bitr4i 290 . . . . . 6 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}))
4135, 40syl6ibr 240 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4241exlimdv 1846 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
431, 42syl5bi 230 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4443con4d 112 . 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 6374 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝐶𝐴𝑥𝐶) → (𝐻𝑥) ∈ (𝐻𝐶))
47463expia 1258 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝐶𝐴) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4847adantrr 748 . . . . . . . . 9 ((𝐻 Fn 𝐴 ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4945, 48sylan 486 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
5049adantrd 482 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝐻𝐶)))
5127biimpd 217 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥)𝑆(𝐻𝐷)))
52 fvex 6094 . . . . . . . . . . . . . . . 16 (𝐻𝑥) ∈ V
5352eliniseg 5396 . . . . . . . . . . . . . . 15 ((𝐻𝐷) ∈ V → ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
5414, 53ax-mp 5 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷))
5551, 54syl6ibr 240 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5626, 55sylbid 228 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5756exp32 628 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))))
582, 57syl9r 75 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝑥𝐶 → (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))))
5958com34 88 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐶𝐴 → (𝐷𝐴 → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))))
6059imp32 447 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
6160impd 445 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6250, 61jcad 553 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
63 elin 3753 . . . . . 6 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6462, 36, 633imtr4g 283 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)}))))
65 n0i 3874 . . . . 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 1846 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6838, 67syl5bi 230 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6944, 68impcon4bid 215 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 194  wa 382   = wceq 1474  wex 1694  wcel 1975  wrex 2892  Vcvv 3168  cin 3534  wss 3535  c0 3869  {csn 4120   class class class wbr 4573  ccnv 5023  cima 5027   Fn wfn 5781  1-1-ontowf1o 5785  cfv 5786   Isom wiso 5787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-f1o 5793  df-fv 5794  df-isom 5795
This theorem is referenced by:  isofrlem  6464
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