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Theorem isomin 6779
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 4094 . . . 4 (¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})))
2 ssel 3755 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝑥𝐶𝑥𝐴))
3 isof1o 6765 . . . . . . . . . . . . . . 15 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofn 6321 . . . . . . . . . . . . . . 15 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
5 fnbrfvb 6424 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
65ex 401 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . 12 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐶) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
109rexbidva 3196 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (∃𝑥𝐶 (𝐻𝑥) = 𝑦 ↔ ∃𝑥𝐶 𝑥𝐻𝑦))
11 vex 3353 . . . . . . . . . . . 12 𝑦 ∈ V
1211elima 5653 . . . . . . . . . . 11 (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 𝑥𝐻𝑦)
1310, 12syl6rbbr 281 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ (𝐻𝐶) ↔ ∃𝑥𝐶 (𝐻𝑥) = 𝑦))
14 fvex 6388 . . . . . . . . . . 11 (𝐻𝐷) ∈ V
1511eliniseg 5676 . . . . . . . . . . 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 624 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → ((𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3958 . . . . . . . . 9 (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦 ∈ (𝐻𝐶) ∧ 𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
19 r19.41v 3236 . . . . . . . . 9 (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐶 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
2017, 18, 193bitr4g 305 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐶𝐴) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
2120adantrr 708 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
22 breq1 4812 . . . . . . . . . . . . . 14 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
2322biimpar 469 . . . . . . . . . . . . 13 (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → (𝐻𝑥)𝑆(𝐻𝐷))
24 vex 3353 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
2524eliniseg 5676 . . . . . . . . . . . . . . 15 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2625ad2antll 720 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
27 isorel 6768 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2826, 27bitrd 270 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
2923, 28syl5ibr 237 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷})))
3029exp32 411 . . . . . . . . . . 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 409 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → 𝑥 ∈ (𝑅 “ {𝐷}))))
3433reximdvai 3161 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥𝐶 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
3521, 34sylbid 231 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷})))
36 elin 3958 . . . . . . . 8 (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
3736exbii 1943 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
38 neq0 4094 . . . . . . 7 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})))
39 df-rex 3061 . . . . . . 7 (∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}) ↔ ∃𝑥(𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})))
4037, 38, 393bitr4i 294 . . . . . 6 (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ∃𝑥𝐶 𝑥 ∈ (𝑅 “ {𝐷}))
4135, 40syl6ibr 243 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
4241exlimdv 2028 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑦 𝑦 ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) → ¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅))
431, 42syl5bi 233 . . 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)})) = ∅))
453, 4syl 17 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
46 fnfvima 6689 . . . . . . . . . . 11 ((𝐻 Fn 𝐴𝐶𝐴𝑥𝐶) → (𝐻𝑥) ∈ (𝐻𝐶))
47463expia 1150 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝐶𝐴) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4847adantrr 708 . . . . . . . . 9 ((𝐻 Fn 𝐴 ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
4945, 48sylan 575 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝐻𝑥) ∈ (𝐻𝐶)))
5049adantrd 485 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝐻𝐶)))
5127biimpd 220 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥)𝑆(𝐻𝐷)))
52 fvex 6388 . . . . . . . . . . . . . . . 16 (𝐻𝑥) ∈ V
5352eliniseg 5676 . . . . . . . . . . . . . . 15 ((𝐻𝐷) ∈ V → ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
5414, 53ax-mp 5 . . . . . . . . . . . . . 14 ((𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}) ↔ (𝐻𝑥)𝑆(𝐻𝐷))
5551, 54syl6ibr 243 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5626, 55sylbid 231 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
5756exp32 411 . . . . . . . . . . 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 409 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥𝐶 → (𝑥 ∈ (𝑅 “ {𝐷}) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
6160impd 398 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6250, 61jcad 508 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝑥𝐶𝑥 ∈ (𝑅 “ {𝐷})) → ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)}))))
63 elin 3958 . . . . . 6 ((𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ((𝐻𝑥) ∈ (𝐻𝐶) ∧ (𝐻𝑥) ∈ (𝑆 “ {(𝐻𝐷)})))
6462, 36, 633imtr4g 287 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → (𝐻𝑥) ∈ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)}))))
65 n0i 4084 . . . . 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 2028 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (∃𝑥 𝑥 ∈ (𝐶 ∩ (𝑅 “ {𝐷})) → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6838, 67syl5bi 233 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐶 ∩ (𝑅 “ {𝐷})) = ∅ → ¬ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
6944, 68impcon4bid 218 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  cin 3731  wss 3732  c0 4079  {csn 4334   class class class wbr 4809  ccnv 5276  cima 5280   Fn wfn 6063  1-1-ontowf1o 6067  cfv 6068   Isom wiso 6069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-f1o 6075  df-fv 6076  df-isom 6077
This theorem is referenced by:  isofrlem  6782
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